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9.7: Use Properties of Rectangles, Triangles, and Trapezoids (Part 2) - Mathematics


Use the Properties of Triangles

We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle in Figure (PageIndex{9}), we’ve labeled the length b and the width h, so it’s area is bh.

Figure (PageIndex{9}) - The area of a rectangle is the base, b, times the height, h.

We can divide this rectangle into two congruent triangles (Figure (PageIndex{10})). Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or (dfrac{1}{2})bh. This example helps us see why the formula for the area of a triangle is A = (dfrac{1}{2})bh.

Figure (PageIndex{10}) - A rectangle can be divided into two triangles of equal area. The area of each triangle is one-half the area of the rectangle.

The formula for the area of a triangle is A = (dfrac{1}{2})bh, where b is the base and h is the height. To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a 90° angle with the base. Figure (PageIndex{11}) shows three triangles with the base and height of each marked.

Figure (PageIndex{11}) - The height h of a triangle is the length of a line segment that connects the the base to the opposite vertex and makes a 90° angle with the base.

Definition: Triangle Properties

For any triangle ΔABC, the sum of the measures of the angles is 180°.$$m angle A + m angle B + m angle C = 180°$$The perimeter of a triangle is the sum of the lengths of the sides.$$P = a + b + c$$The area of a triangle is one-half the base, b, times the height, h.$$A = dfrac{1}{2} bh]

Example (PageIndex{9}):

Find the area of a triangle whose base is 11 inches and whose height is 8 inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.the area of the triangle
Step 3. Name. Choose a variable to represent it.let A = area of the triangle
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.A = 44 square inches
Step 6. Check.$$egin{split} A &= dfrac{1}{2} bh 44 &stackrel{?}{=} dfrac{1}{2} (11)8 44 &= 44; checkmark end{split}$$
Step 7. Answer the question.The area is 44 square inches.

Exercise (PageIndex{17}):

Find the area of a triangle with base 13 inches and height 2 inches.

Answer

13 sq. in.

Exercise (PageIndex{18}):

Find the area of a triangle with base 14 inches and height 7 inches.

Answer

49 sq. in.

Example (PageIndex{10}):

The perimeter of a triangular garden is 24 feet. The lengths of two sides are 4 feet and 9 feet. How long is the third side?

Solution

Step 1. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.length of the third side of a triangle
Step 3. Choose a variable to represent it.Let c = the third side
Step 4.Translate. Substitute in the given information.
Step 5. Solve the equation.$$egin{split} 24 &= 13 + c 11 &= c end{split}$$
Step 6. Check.$$egin{split} P &= a + b + c 24 &stackrel{?}{=} 4 + 9 + 11 24 &= 24; checkmark end{split}$$
Step 7. Answer the question.The third side is 11 feet long.

Exercise (PageIndex{19}):

The perimeter of a triangular garden is 48 feet. The lengths of two sides are 18 feet and 22 feet. How long is the third side?

Answer

8 ft

Exercise (PageIndex{20}):

The lengths of two sides of a triangular window are 7 feet and 5 feet. The perimeter is 18 feet. How long is the third side?

Answer

6 ft

Example (PageIndex{11}):

The area of a triangular church window is 90 square meters. The base of the window is 15 meters. What is the window’s height?

Solution

Step 1. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.height of a triangle
Step 3. Choose a variable to represent it.Let h = the height
Step 4.Translate. Substitute in the given information.
Step 5. Solve the equation.$$egin{split} 90 &= dfrac{15}{2} h 12 &= h end{split}$$
Step 6. Check.$$egin{split} A &= dfrac{1}{2} bh 90 &stackrel{?}{=} dfrac{1}{2} cdot 15 cdot 12 90 &= 90; checkmark end{split}$$
Step 7. Answer the question.The height of the triangle is 12 meters.

Exercise (PageIndex{21}):

The area of a triangular painting is 126 square inches. The base is 18 inches. What is the height?

Answer

14 in.

Exercise (PageIndex{22}):

A triangular tent door has an area of 15 square feet. The height is 5 feet. What is the base?

Answer

6 ft

Isosceles and Equilateral Triangles

Besides the right triangle, some other triangles have special names. A triangle with two sides of equal length is called an isosceles triangle. A triangle that has three sides of equal length is called an equilateral triangle. Figure (PageIndex{12}) shows both types of triangles.

Figure (PageIndex{12}) - In an isosceles triangle, two sides have the same length, and the third side is the base. In an equilateral triangle, all three sides have the same length.

Definition: Isosceles and Equilateral Triangles

An isosceles triangle has two sides the same length.

An equilateral triangle has three sides of equal length.

Example (PageIndex{12}):

The perimeter of an equilateral triangle is 93 inches. Find the length of each side.

Solution

Step 1. Draw the figure and label it with the given information.

Perimeter = 93 in.

Step 2. Identify what you are looking for.length of the sides of an equilateral triangle
Step 3. Choose a variable to represent it.Let s = length of each side
Step 4.Translate. Substitute.
Step 5. Solve the equation.$$egin{split} 93 &= 3s 31 &= s end{split}$$
Step 6. Check.$$egin{split} 93 &= 31 + 31 + 31 93 &= 93; checkmark end{split}$$
Step 7. Answer the question.Each side is 31 inches.

Exercise (PageIndex{23}):

Find the length of each side of an equilateral triangle with perimeter 39 inches.

Answer

13 in.

Exercise (PageIndex{24}):

Find the length of each side of an equilateral triangle with perimeter 51 centimeters.

Answer

17 cm

Example (PageIndex{13}):

Arianna has 156 inches of beading to use as trim around a scarf. The scarf will be an isosceles triangle with a base of 60 inches. How long can she make the two equal sides?

Solution

Step 1. Draw the figure and label it with the given information.

P = 156 in.

Step 2. Identify what you are looking for.the lengths of the two equal sides
Step 3. Choose a variable to represent it.Let s = the length of each side
Step 4.Translate. Substitute in the given information.
Step 5. Solve the equation.$$egin{split} 156 &= 2s + 60 96 &= 2s 48 &= s end{split}$$
Step 6. Check.$$egin{split} p &= a + b + c 156 &stackrel{?}{=} 48 + 60 + 48 156 &= 156; checkmark end{split}$$
Step 7. Answer the question.Arianna can make each of the two equal sides 48 inches long.

Exercise (PageIndex{25}):

A backyard deck is in the shape of an isosceles triangle with a base of 20 feet. The perimeter of the deck is 48 feet. How long is each of the equal sides of the deck?

Answer

14 ft

Exercise (PageIndex{26}):

A boat’s sail is an isosceles triangle with base of 8 meters. The perimeter is 22 meters. How long is each of the equal sides of the sail?

Answer

7 m

Use the Properties of Trapezoids

A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base b, and the length of the bigger base B. The height, h, of a trapezoid is the distance between the two bases as shown in Figure (PageIndex{13}).

Figure (PageIndex{13}) - A trapezoid has a larger base, B, and a smaller base, b. The height h is the distance between the bases.

The formula for the area of a trapezoid is:

[Area_{trapezoid} = dfrac{1}{2} h(b + B)]

Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. See Figure (PageIndex{14}).

Figure (PageIndex{14}) - Splitting a trapezoid into two triangles may help you understand the formula for its area.

The height of the trapezoid is also the height of each of the two triangles. See Figure (PageIndex{15}).

Figure (PageIndex{15})

The formula for the area of a trapezoid is

[Area_{trapezoid} = dfrac{1}{2} h ( extcolor{blue}{b} + extcolor{red}{B})]

If we distribute, we get,

Definition: Properties of Trapezoids

  • A trapezoid has four sides. See Figure 9.25.
  • Two of its sides are parallel and two sides are not.
  • The area, A, of a trapezoid is A = (dfrac{1}{2})h(b + B).

Example (PageIndex{14}):

Find the area of a trapezoid whose height is 6 inches and whose bases are 14 and 11 inches.

Solution

Step 1. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.the area of the trapezoid
Step 3. Choose a variable to represent it.Let A = the area
Step 4.Translate. Substitute.
Step 5. Solve the equation.$$egin{split} A &= dfrac{1}{2} cdot 6(25) A &= 3(25) A &= 75; square; inches end{split}$$
Step 6. Check: Is this answer reasonable?

If we draw a rectangle around the trapezoid that has the same big base B and a height h, its area should be greater than that of the trapezoid.

If we draw a rectangle inside the trapezoid that has the same little base b and a height h, its area should be smaller than that of the trapezoid.

The area of the larger rectangle is 84 square inches and the area of the smaller rectangle is 66 square inches. So it makes sense that the area of the trapezoid is between 84 and 66 square inches

Step 7. Answer the question.The area of the trapezoid is 75 square inches.

Exercise (PageIndex{27}):

The height of a trapezoid is 14 yards and the bases are 7 and 16 yards. What is the area?

Answer

161 sq. yd

Exercise (PageIndex{28}):

The height of a trapezoid is 18 centimeters and the bases are 17 and 8 centimeters. What is the area?

Answer

255 sq. cm

Example (PageIndex{15}):

Find the area of a trapezoid whose height is 5 feet and whose bases are 10.3 and 13.7 feet.

Solution

Step 1. Draw the figure and label it with the given information.
Step 2. Substitute.
Step 5. Solve the equation.$$egin{split} A &= dfrac{1}{2} cdot 5(24) A &= 12 cdot 5 A &= 60; square; feet end{split}$$
Step 6. Check: Is this answer reasonable? The area of the trapezoid should be less than the area of a rectangle with base 13.7 and height 5, but more than the area of a rectangle with base 10.3 and height 5.
Step 7. Answer the question.The area of the trapezoid is 60 square feet.

Exercise (PageIndex{29}):

The height of a trapezoid is 7 centimeters and the bases are 4.6 and 7.4 centimeters. What is the area?

Answer

42 sq. cm

Exercise (PageIndex{30}):

The height of a trapezoid is 9 meters and the bases are 6.2 and 7.8 meters. What is the area?

Answer

63 sq. m

Example (PageIndex{16}):

Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of 3.4 yards and the bases are 8.2 and 5.6 yards. How many square yards will be available to plant?

Solution

Step 1. Draw the figure and label it with the given information.
Step 2. Substitute.
Step 5. Solve the equation.$$egin{split} A &= dfrac{1}{2} cdot (3.4)(13.8) A &= 23.46; square; yards end{split}$$

Step 6. Check: Is this answer reasonable? Yes. The area of the trapezoid is less than the area of a rectangle with a base of 8.2 yd and height 3.4 yd, but more than the area of a rectangle with base 5.6 yd and height 3.4 yd.

Step 7. Answer the question.Vinny has 23.46 square yards in which he can plant.

Exercise (PageIndex{31}):

Lin wants to sod his lawn, which is shaped like a trapezoid. The bases are 10.8 yards and 6.7 yards, and the height is 4.6 yards. How many square yards of sod does he need?

Answer

40.25 sq. yd

Exercise (PageIndex{32}):

Kira wants cover his patio with concrete pavers. If the patio is shaped like a trapezoid whose bases are 18 feet and 14 feet and whose height is 15 feet, how many square feet of pavers will he need?

Answer

240 sq. ft

Practice Makes Perfect

Understand Linear, Square, and Cubic Measure

In the following exercises, determine whether you would measure each item using linear, square, or cubic units.

  1. amount of water in a fish tank
  2. length of dental floss
  3. living area of an apartment
  4. floor space of a bathroom tile
  5. height of a doorway
  6. capacity of a truck trailer

In the following exercises, find the (a) perimeter and (b) area of each figure. Assume each side of the square is 1 cm.

Use the Properties of Rectangles

In the following exercises, find the (a) perimeter and (b) area of each rectangle.

  1. The length of a rectangle is 85 feet and the width is 45 feet.
  2. The length of a rectangle is 26 inches and the width is 58 inches.
  3. A rectangular room is 15 feet wide by 14 feet long.
  4. A driveway is in the shape of a rectangle 20 feet wide by 35 feet long.

In the following exercises, solve.

  1. Find the length of a rectangle with perimeter 124 inches and width 38 inches.
  2. Find the length of a rectangle with perimeter 20.2 yards and width of 7.8 yards.
  3. Find the width of a rectangle with perimeter 92 meters and length 19 meters.
  4. Find the width of a rectangle with perimeter 16.2 meters and length 3.2 meters.
  5. The area of a rectangle is 414 square meters. The length is 18 meters. What is the width?
  6. The area of a rectangle is 782 square centimeters. The width is 17 centimeters. What is the length?
  7. The length of a rectangle is 9 inches more than the width. The perimeter is 46 inches. Find the length and the width.
  8. The width of a rectangle is 8 inches more than the length. The perimeter is 52 inches. Find the length and the width.
  9. The perimeter of a rectangle is 58 meters. The width of the rectangle is 5 meters less than the length. Find the length and the width of the rectangle.
  10. The perimeter of a rectangle is 62 feet. The width is 7 feet less than the length. Find the length and the width.
  11. The width of the rectangle is 0.7 meters less than the length. The perimeter of a rectangle is 52.6 meters. Find the dimensions of the rectangle.
  12. The length of the rectangle is 1.1 meters less than the width. The perimeter of a rectangle is 49.4 meters. Find the dimensions of the rectangle.
  13. The perimeter of a rectangle of 150 feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.
  14. The length of a rectangle is three times the width. The perimeter is 72 feet. Find the length and width of the rectangle.
  15. The length of a rectangle is 3 meters less than twice the width. The perimeter is 36 meters. Find the length and width.
  16. The length of a rectangle is 5 inches more than twice the width. The perimeter is 34 inches. Find the length and width.
  17. The width of a rectangular window is 24 inches. The area is 624 square inches. What is the length?
  18. The length of a rectangular poster is 28 inches. The area is 1316 square inches. What is the width?
  19. The area of a rectangular roof is 2310 square meters. The length is 42 meters. What is the width?
  20. The area of a rectangular tarp is 132 square feet. The width is 12 feet. What is the length?
  21. The perimeter of a rectangular courtyard is 160 feet. The length is 10 feet more than the width. Find the length and the width.
  22. The perimeter of a rectangular painting is 306 centimeters. The length is 17 centimeters more than the width. Find the length and the width.
  23. The width of a rectangular window is 40 inches less than the height. The perimeter of the doorway is 224 inches. Find the length and the width.
  24. The width of a rectangular playground is 7 meters less than the length. The perimeter of the playground is 46 meters. Find the length and the width.

Use the Properties of Triangles

In the following exercises, solve using the properties of triangles.

  1. Find the area of a triangle with base 12 inches and height 5 inches.
  2. Find the area of a triangle with base 45 centimeters and height 30 centimeters.
  3. Find the area of a triangle with base 8.3 meters and height 6.1 meters.
  4. Find the area of a triangle with base 24.2 feet and height 20.5 feet.
  5. A triangular flag has base of 1 foot and height of 1.5 feet. What is its area?
  6. A triangular window has base of 8 feet and height of 6 feet. What is its area?
  7. If a triangle has sides of 6 feet and 9 feet and the perimeter is 23 feet, how long is the third side?
  8. If a triangle has sides of 14 centimeters and 18 centimeters and the perimeter is 49 centimeters, how long is the third side?
  9. What is the base of a triangle with an area of 207 square inches and height of 18 inches?
  10. What is the height of a triangle with an area of 893 square inches and base of 38 inches?
  11. The perimeter of a triangular reflecting pool is 36 yards. The lengths of two sides are 10 yards and 15 yards. How long is the third side?
  12. A triangular courtyard has perimeter of 120 meters. The lengths of two sides are 30 meters and 50 meters. How long is the third side?
  13. An isosceles triangle has a base of 20 centimeters. If the perimeter is 76 centimeters, find the length of each of the other sides.
  14. An isosceles triangle has a base of 25 inches. If the perimeter is 95 inches, find the length of each of the other sides.
  15. Find the length of each side of an equilateral triangle with a perimeter of 51 yards.
  16. Find the length of each side of an equilateral triangle with a perimeter of 54 meters.
  17. The perimeter of an equilateral triangle is 18 meters. Find the length of each side.
  18. The perimeter of an equilateral triangle is 42 miles. Find the length of each side.
  19. The perimeter of an isosceles triangle is 42 feet. The length of the shortest side is 12 feet. Find the length of the other two sides.
  20. The perimeter of an isosceles triangle is 83 inches. The length of the shortest side is 24 inches. Find the length of the other two sides.
  21. A dish is in the shape of an equilateral triangle. Each side is 8 inches long. Find the perimeter.
  22. A floor tile is in the shape of an equilateral triangle. Each side is 1.5 feet long. Find the perimeter.
  23. A road sign in the shape of an isosceles triangle has a base of 36 inches. If the perimeter is 91 inches, find the length of each of the other sides.
  24. A scarf in the shape of an isosceles triangle has a base of 0.75 meters. If the perimeter is 2 meters, find the length of each of the other sides.
  25. The perimeter of a triangle is 39 feet. One side of the triangle is 1 foot longer than the second side. The third side is 2 feet longer than the second side. Find the length of each side.
  26. The perimeter of a triangle is 35 feet. One side of the triangle is 5 feet longer than the second side. The third side is 3 feet longer than the second side. Find the length of each side.
  27. One side of a triangle is twice the smallest side. The third side is 5 feet more than the shortest side. The perimeter is 17 feet. Find the lengths of all three sides.
  28. One side of a triangle is three times the smallest side. The third side is 3 feet more than the shortest side. The perimeter is 13 feet. Find the lengths of all three sides.

Use the Properties of Trapezoids

In the following exercises, solve using the properties of trapezoids.

  1. The height of a trapezoid is 12 feet and the bases are 9 and 15 feet. What is the area?
  2. The height of a trapezoid is 24 yards and the bases are 18 and 30 yards. What is the area?
  3. Find the area of a trapezoid with a height of 51 meters and bases of 43 and 67 meters.
  4. Find the area of a trapezoid with a height of 62 inches and bases of 58 and 75 inches.
  5. The height of a trapezoid is 15 centimeters and the bases are 12.5 and 18.3 centimeters. What is the area?
  6. The height of a trapezoid is 48 feet and the bases are 38.6 and 60.2 feet. What is the area?
  7. Find the area of a trapezoid with a height of 4.2 meters and bases of 8.1 and 5.5 meters.
  8. Find the area of a trapezoid with a height of 32.5 centimeters and bases of 54.6 and 41.4 centimeters.
  9. Laurel is making a banner shaped like a trapezoid. The height of the banner is 3 feet and the bases are 4 and 5 feet. What is the area of the banner?
  10. Niko wants to tile the floor of his bathroom. The floor is shaped like a trapezoid with width 5 feet and lengths 5 feet and 8 feet. What is the area of the floor?
  11. Theresa needs a new top for her kitchen counter. The counter is shaped like a trapezoid with width 18.5 inches and lengths 62 and 50 inches. What is the area of the counter?
  12. Elena is knitting a scarf. The scarf will be shaped like a trapezoid with width 8 inches and lengths 48.2 inches and 56.2 inches. What is the area of the scarf?

Everyday Math

  1. Fence Jose just removed the children’s playset from his back yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a 50 foot roll of fence in his garage that he plans to use. To fit in the backyard, the width of the garden must be 10 feet. How long can he make the other side if he wants to use the entire roll of fence?
  2. Gardening Lupita wants to fence in her tomato garden. The garden is rectangular and the length is twice the width. It will take 48 feet of fencing to enclose the garden. Find the length and width of her garden.
  3. Fence Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are 6 feet, 8 feet, and 10 feet. The fence costs $10 per foot. How much will it cost for Christa to fence in her flowerbed?
  4. Painting Caleb wants to paint one wall of his attic. The wall is shaped like a trapezoid with height 8 feet and bases 20 feet and 12 feet. The cost of the painting one square foot of wall is about $0.05. About how much will it cost for Caleb to paint the attic wall?

Writing Exercises

  1. If you need to put tile on your kitchen floor, do you need to know the perimeter or the area of the kitchen? Explain your reasoning.
  2. If you need to put a fence around your backyard, do you need to know the perimeter or the area of the backyard? Explain your reasoning.
  3. Look at the two figures. (a) Which figure looks like it has the larger area? Which looks like it has the larger perimeter? (b) Now calculate the area and perimeter of each figure. Which has the larger area? Which has the larger perimeter?

  1. The length of a rectangle is 5 feet more than the width. The area is 50 square feet. Find the length and the width. (a) Write the equation you would use to solve the problem. (b) Why can’t you solve this equation with the methods you learned in the previous chapter?

Self Check

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

(b) On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


Why I hate the definition of trapezoids (part 3)

Yes it’s true. I’m writing about trapezoids again (having written passionately about them here and here previously). I’ve been taking a break from blogging, as I usually do in the summer. For us, school starts in just two weeks. So I thought I’d come out of my shell and post something…and of course I always have something to say about trapezoids :-).

Let’s start with the following easy test question. Don’t peek. See if you can answer the question without any help.

Which of the following quadrilaterals are trapezoids?

Before giving the answer, let me first just remind you about my very strongly held position. I believe that instead of this typical textbook definition (the “exclusive definition” we’ll call it) that reads:

“A quadrilateral with one and only one pair of parallel sides.”

the definition should be made inclusive, and read:

“A quadrilateral with at least one pair of parallel sides.”

So the test question above was easy, right? Quadrilaterals (A) and (C) are trapezoids, I hear you say.

Not so fast!! If you’re using the inclusive definition, then the correct answers are actually (A), (B), (C), (D), and (E). But it gets better: If you were using the the exclusive definition, then NONE of these are trapezoids. In order for (A) and (C) to be trapezoids, under the exclusive definition, you must prove that two sides are parallel AND the two remaining sides are not parallel (and you can’t assume that from the picture…especially for (C)!).

Can you see the absurdity of the exclusive definition now?

I finish by offering the following list of reasons why the inclusive definition is better (can you suggest more reasons?):

  1. All other quadrilaterals are defined in the inclusive way, so that quadrilaterals “beneath” them inherit all the properties of their “parents.” A square is a rectangle because a square meets the definition of a rectangle. Likewise, parallelograms, rectangles, rhombuses, and squares should all be special cases of a trapezoid.
  2. The area formula for a trapezoid still works, even if the legs are parallel. It’s true! The area formula works fine for a parallelogram, rectangle, rhombus, or square.
  3. No other definitions break when you use the inclusive definition. With the exception of the definition that some texts use for an isosceles trapezoid. Those texts define an isosceles trapezoid has having both legs congruent, which would make a parallelogram an isosceles trapezoid. Instead, define an isosceles trapezoid as having base angles congruent, or equivalently, having a line of symmetry.
  4. The trapezoidal approximation method in Calculus doesn’t fail when one of the trapezoids is actually a rectangle. But under the exclusive definition, you would have to change its name to the “trapezoidal and/or rectangular approximation method,” or perhaps ban people from doing the trapezoidal method on problems like this one: Approximate using the trapezoidal method with 5 equal intervals. (Note here that the center trapezoid is actually a rectangle…God forbid!!)
  5. When proving that a quadrilateral is a trapezoid, one can stop after proving just two sides are parallel. But with the exclusive definition, in order to prove that a quadrilateral is a trapezoid, you would have to prove two sides are parallel AND the other two sides are not parallel (see the beginning of this post!).

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Primary Resource Type: Original Tutorial

Attachments

  • MAFS.5.G.2.3: Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

Explore the defining attributes of trapezoids--a special type of quadrilateral--and classify them using diagrams in this interactive tutorial. You'll also learn how two different definitions for a trapezoid can change affect classifications of quadrilaterals.

This part 6 in a 6-part series. Click below to explore the other tutorials in the series.


Geometry, Common Core Style

Section 5-2 of the U of Chicago text covers the various types of quadrilaterals. There are no theorems in this section, but just definitions. The concept of definition is important to the study of geometry, and in no lesson so far are definitions more prominent than in this lesson.

The lesson begins by defining parallelogram, rhombus, rectangle, and square. There's nothing wrong with any of those definitions. But then we reach a controversial definition -- that of trapezoid:

Definition:
A quadrilateral is a trapezoid if and only if it has at least one pair of parallel sides.
(emphasis mine)

Just as with the definition of parallel back in Section 1-7, we have two extra words that distinguish this from a traditional definition of trapezoid -- "at least." In other textbooks, no parallelogram is a trapezoid, but in the U of Chicago text, every parallelogram is a trapezoid!

To understand what's going on here, let's go back to the first geometer who defined some of the terms in the quadrilateral hierarchy -- of course, I'm talking about Euclid:

Of quadrilateral figures, a square is that which is both equilateral and right-angled an oblong that which is right-angled but not equilateral a rhombus that which is equilateral but not right-angled and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

Of course, the modern term for "oblong" is rectangle, and a "rhomboid" is now a parallelogram. The word "trapezia" is actually plural of "trapezium." In British English, a "trapezium" is what we Americans would call a trapezoid, but to Euclid, any quadrilateral that is not a parallelogram (or below on the quadrilateral hierarchy) is a "trapezium." But the important part here is that to Euclid, a square, for example, is neither a rectangle (oblong) nor a rhombus. He makes sure to say that a rectangle (oblong) is "not equilateral," and that a rhombus is "not right-angled." And of course, neither a rectangle nor a rhombus is a parallelogram (rhomboid).

These are called "exclusive" definitions. For Euclid, there was no quadrilateral hierarchy -- each class of quadrilaterals was disjoint from the others. But since the days of Euclid, more and more geometry texts have slowly added more "inclusive" definitions.

One of the first inclusive definitions I've seen was the definition of rectangle. It was mentioned on an episode of Square One TV, when a Pacman parody character named Mathman was supposed to eat rectangles, and then ate a square because "every square's a rectangle." I would provide a YouTube link, but I haven't found the link in years and it doesn't come up in a search. (I even remember someone posting in the comments that just as for me, his first encounter of the inclusive definition of rectangle was through watching that clip when it first aired so many years ago!)

But many of my family members were also teachers, and one relative gave me an old textbook that still mentioned some exclusive definitions. In particular, it declared that a square isn't a rhombus. A little later, my fifth grade teacher then taught the inclusive definition of rhombus. I then blurted out that a square isn't a rhombus, then actually brought the old text to school to prove it! She replied, "Wow!" but then, if I recalled correctly, told me that this definition was old, and that by the new definition, a square is a rhombus. And so all modern texts classify the square as both a rectangle and a rhombus, and that all of these are considered parallelograms.

So we see that there is a tendency for definitions to grow more inclusive as time goes on. (We see this happening in politics as well -- for example, the definition of marriage. But I digress.) And so we see the next natural step is for the parallelogram to be considered a trapezoid.

One of the first advocates I saw for an inclusive definition of trapezoid is the famous Princeton mathematician John H. Conway. He is best known for inventing the mathematical Game of Life, which has its own website:

But Conway also specializes in other fields of mathematics, such as geometry and group theory (which is, in some ways, the study of symmetry). Twelve years ago, he posted the following information about why he prefers inclusive definitions:

The preference for exclusive definitions arises, I think, from
what I call "the descriptive use". Of course, one wouldn't DESCRIBE
a square table as "rectangular", since that would wantonly use
a longer term to convey less information. So in descriptive uses,
there's a natural presumption that a table called "rectangular"
won't in fact be square - in other words, a natural presumption
that the terms will be used exclusively.

But the descriptive use is unimportant to geometry, where the
really important thing is the truth of theorems. This means that we
should use a term "A" to include "B" if all the identities that
hold for all "A"s will also hold for all "B"s (in the way that
the trapezoid area theorem holds for all parallelograms, for
instance).

You might worry about the consistency of switching to the
inclusive use while other people continue with the exclusive one.
But there can be no consistency with people who are inconsistent!
I've seen many geometry books that MAKE the exclusive definitions,
but none that manage to USE them consistently for more than a few
theorems.

Indeed, Conway advocated taking it one step forward and actually abolishing the trapezoid and having only the isosceles trapezoid in the hierarchy! After all, there's not much one can say about a trapezoid that's not isosceles -- just look ahead to Section 5-5. There's only one theorem listed there about general trapezoids -- the Trapezoid Angle Theorem, and that's really just the Same-Side Interior Angle Consequence Theorem that can be proved without reference to trapezoids at all. All the other theorems in the lesson refer to isosceles trapezoids. In particular, the symmetry theorems in the lesson refer to isosceles trapezoids. (Recall that Conway's specializes in group theory -- which as I wrote above is the study of symmetry.) I suspect that the only reason that we have general trapezoids is that they are the simplest quadrilateral for which an area formula can be given.

This is now another digression from Common Core Geometry, so I'll just provide another link. Notice that here, Conway also proposes a hexagon hierarchy based on symmetry. There's also a pentagon hierarchy, but there are only three types of pentagons -- general, line-symmetric, and regular -- just as there are for triangles. It's easier to make figures with an even number of sides symmetric.

Another advocate of inclusive definitions is Mr. Chase, a high school math teacher from Maryland. I see that he is so passionate about the inclusive definition of trapezoid that he devoted three whole blog posts to why he hates the exclusive definition of trapezoid:

One reason Chase states for using inclusive definitions is that it simplifies proofs:

When proving that a quadrilateral is a trapezoid, one can stop after proving just two sides are parallel . But with the exclusive definition, in order to prove that a quadrilateral is a trapezoid, you would have to prove two sides are parallel AND the other two sides are not parallel.

Regarding some of the others to whom I refer regularly, Dr. Wu uses the inclusive definition:

"A quadrilateral with at least one pair of opposite sides that are parallel is called a trapezoid. A trapezoid with two pairs of parallel opposite sides is called a parallelogram."

while Dr. Mason uses the exclusive definition:

"A trapezoid is by definition a quadrilateral with precisely one pair of parallel sides."
(emphasis Dr. M's)

So which definition should I use for trapezoid? Well, this is a Common Core blog, so the definition favored by Common Core should have priority over all other definitions. The following is a link to the information that will appear on the PARCC End-of-Year Assessment for geometry:

And right there in the column under "Clarifications," it reads:

i) A trapezoid is defined as “A quadrilateral with at least one pair of parallel sides.”

And that plainly settles it. The PARCC Common Core assessment uses the inclusive definition of trapezoid, and so it's my duty on a Common Core blog to use the Common Core definition. Of course, we notice that this is the definition given by PARCC -- but so far I've seen no information on what definition Smarter Balanced is using. It would be tragic if PARCC were to use one definition and Smarter Balanced the other. But as I can't say anything about Smarter Balanced, I will use the only definition that's known to be on a Common Core test, and that's the inclusive definition. The fact that this definition is already used by the U of Chicago is icing on the cake.

There is one problem with the inclusive definition of trapezoid, and that's when we try to define isosceles trapezoid. The word isosceles suggests that, just as in an isosceles triangle, an isosceles trapezoid has two equal sides -- the sides adjacent to the (parallel) bases. But in a parallelogram, where either pair of opposite sides can be considered the bases -- the sides adjacent to these bases are also equal. This would make every parallelogram an isosceles trapezoid. But this isn't desirable -- an isosceles trapezoid has several properties that parallelograms in general lack. The diagonals of an isosceles trapezoid are equal, but those of a parallelogram in general aren't. But the diagonals of a rectangle are equal. So we'd like to consider rectangles, but not parallelograms in general, to be isosceles trapezoids.

This dilemma is mentioned in the comments at one of the Chase links. It's pointed out that there are two ways out of this mess -- we may either define isosceles trapezoid in terms of symmetry, as Conway does, or we can use the U of Chicago's definition:

"A trapezoid is isosceles if and only if it has a pair of base angles equal in measure."

Some don't like this definition, because it violates linguistic purity -- the word isosceles comes from Greek, and it means "equal legs," not "equal angles." But as it turns out, it's a small price to pay to make the quadrilateral hierarchy and other theorems work out. And besides, any geometer who calls a nine-sided polygon a nonagon should just shut up about linguistic purity!

There's one definition in this lesson that I've neglected to mention -- the kite. As it turns out, there are two definitions of kite, one exclusive and one inclusive. The inclusive definition makes every rhombus (and therefore every square) a kite. Naturally, those who prefer the exclusive definition of trapezoid, like Dr. M, also prefers the exclusive definition of kite, while others take inclusive definitions of both trapezoid and kite. (Wu is silent on this issue -- he doesn't mention kites on his site at all.)

The fact that exclusive definitions make proofs longer -- as mentioned by both Conway and Chase -- is noticeable when we look at Dr. M's lesson on kites (Lesson 6.6 on his site). Since he is using the exclusive definition of kite, Dr. M must make sure that every property that a kite has, such as having a pair of equal opposite angles, applies only to one pair of angles and not to the other -- otherwise the figure would be a parallelogram (indeed a rhombus) and not a kite. But if we were to use the inclusive definition, we don't need to fear that the figure is a rhombus because a rhombus is still considered to be a kite. Dr. M's lesson contains 15 PowerPoint pages, but we could cut out almost half of them simply by using the inclusive definition -- five pages of indirect "not the other" proofs, and two more pages to explain why the "not the other" proofs are needed!

So here is the U of Chicago definition of kite:

A quadrilateral is a kite if and only if it has two distinct pairs of consecutive sides of the same length.

(Notice that here distinct means that there are two different pairs with the same length, for a total of four sides -- not that the lengths themselves must be distinct.)

The text has the quadrilateral hierarchy, but with one link missing, from rhombus to parallelogram. It states that this will be proved in Lesson 5-4 (actually 5-6), since it uses the Alternate Interior Angles Test that doesn't appear until that lesson. But here on this blog, we already proved the AIA Test, so we can actually prove the entire hierarchy right now.

Now some might notice something here. Yesterday, I wrote that the first four sections of Chapter 5 don't require a Parallel Postulate. Yet today, I'm discussing quadrilaterals like rectangles and squares, and as it turns out, rectangles (and hence squares) don't even exist without a Parallel Postulate!

So what gives here? Actually, every statement proved in this lesson is still true in non-Euclidean geometry, even those like "every rectangle is a parallelogram." If rectangles don't exist, then the statement "every rectangle is a parallelogram" is vacuously true -- there exist zero rectangles, and all zero of them are parallelograms! (Similarly, all unicorns are white.) No statement about rectangles or squares made in this lesson actually requires any of them to exist -- only that if they exist, then they have these properties. Wu does the same trick on his page:

"A quadrilateral all of whose angles are right angles is called a rectangle. A rectangle all of whose sides are of the same length is called a square. Be aware that at this point, we do not know whether there
is a square or not, or worse, whether there is a rectangle or not."

Also, the statement that every rhombus is a parallelogram uses the Alternate Interior Angles Test, but it's the Parallel Consequences, not the Parallel Tests, that require a Parallel Postulate. It is not until Section 5-5, where we derive properties about trapezoids using the consequences of their parallel sides, that we'll need a Parallel Postulate.

Let's move on to the exercises. I decided to throw out the first eight questions since defining, drawing, and placing into a hierarchy the seven types of quadrilaterals fit better in the notes, not in the exercises that come after the notes. As for the other questions, it's interesting to point out how the answers might be different using inclusive/exclusive definitions, or Euclidean/non-Euclidean geometry.

I include the first three true/false questions, from 9-11. Question 9 is true, even in non-Euclidean geometry (where it's vacuously true) and Question 10 is always false. Question 11 is true, but becomes false if we use an exclusive definition of kite. (Notice that my exercises make no reference to trapezoids, but only kites, since kites are coming up sooner in Section 5-4.)

Then I skip to Question 20. If set A is the set of all rectangles and set B is the set of all rhombuses, then A intersect B is the set of all squares. This remains valid even in non-Euclidean geometry, since there set A, the set of all rectangles, becomes the empty set. Then A intersect B would also be the empty set, which equals the set of all squares, since that's the empty set as well.

But there is a similar intersection problem that's too advanced to be given here, and where the answer differs depending on what geometry one is using. The intersection of the set of all parallelograms and the set of all isosceles trapezoids is, in Euclidean geometry, the set of all rectangles. (One way to prove this is to note that isosceles trapezoids have equal diagonals, and -- though this isn't proved in the U of Chicago -- parallelograms with equal diagonals are rectangles.) Yet in hyperbolic geometry, there are no rectangles, but there exists figures that are both parallelograms and isosceles trapezoids -- in particular, the Saccheri quadrilateral is both. A hyperbolic geometer may miss the fact that a Saccheri quadrilateral is an isosceles trapezoid because he/she is using the exclusive definition, where a parallelogram can't be a trapezoid. But in some ways, a Saccheri quadrilateral is more like an Euclidean isosceles trapezoid than a Euclidean parallelogram, since the Saccheri and the isosceles trapezoid share the same type of symmetry line that the general parallelogram lacks.

In Question 21, we prove that NOPQ is a kite -- a proof that requires only four steps (since I always add a Given step to the three steps asked for in the book). But if we use the exclusive definition of kite, NOPQ might not be a kite because it could be a rhombus. Technically speaking, we can't prove that NOPQ is an exclusive kite unless we add another hypothesis, such as circles O and Q having unequal radii.


Geometry, Common Core Style

Today is a student-free day in my new district, similar to October 15th in my old district. The blog calendar follows the old district, and so today is a posting day.

On the other hand, the only way I would sub today is for my old district. Of course, this was very unlikely, and in the end I didn't sub today.

Here I go again, making a traditionalists' post out of schedule. Last week on the scheduled day, I didn't write much and declared that the real traditionalists' posts are the "Sue Teele" posts, as her multiples intelligences are the other side of the debate.

But over the weekend, our main traditionalist Barry Garelick posted. And he's responding to an article written by another author whose ideas we've seen recently -- Jo Boaler:

San Francisco’s Unified School District decided to eliminate access to algebra for 8th graders even if a student is qualified to take such a course. The latest article to justify t he action is one written by Jo Boaler (whose self-styled approach to math education in my opinion and the opinion of many others in education who I respect has been ineffective and damaging) and Alan Schoenfeld, a math professor from UC Berkeley whose stance is consistent with math reformers. I.e., “understanding” takes precedence over procedure, among other things.

Garelick and the other traditionalists have mentioned San Francisco Unified in the past. It's almost always to criticize the district's eighth grade Algebra I policy. He quotes Boaler's article:

“The Common Core State Standards raised the level and rigor of eighth-grade mathematics to include Algebra 1 content as well as geometry and statistical topics previously taught in high school.”

And the traditionalist disagrees. In his article, he ultimately mentions senior-year AP Calculus -- the class that the traditionalists really care about. Raising the rigor of Math 8 to include a little more algebra is irrelevant if it doesn't lead to seniors taking a class called AP Calculus. Oh, and by the way:

Translation: For those students who wish to take calculus in 12th grade, they can double up math courses in 11th grade, so they can take Algebra 2 and Precalculus. As far as what they mean by “conceptually rich courses that benefit everybody”, it’s anybody’s guess.

In other words, it doesn't count as a true path to Calculus unless students can take math for only one period a day, with no summer school, with AP Calculus as the capstone class. (I do agree with Garelick that Algebra II and Pre-Calc together is a very tough load for a junior.)

A high school level course includes rational expressions (i.e., algebraic fractions), polynomial division, factoring, quadratic equations, and direct and inverse variation. The 8th grade standards do not include these. I teach an 8th grade math class as well as high school algebra for 8th graders.

I agree only in part. I usually consider Common Core 8 to line up with the first half of Algebra I. So factoring and quadratics are Algebra 1B topics that don't appear in Common Core 8. As for rational expressions and polynomial long division, these appear in some Algebra I texts, but many high school teachers save these for the end of the year and ultimately skip these topics. Since Garelick states that he teaches eighth Algebra I, I wonder whether he teaches these topics and if he does, what letter grades his eighth graders earn on the tests. (As for direct variation, Garelick addresses this later in his post.)

Also, notice that Boaler never claims that Common Core Math 8 is more rigorous than Algebra I. She means that Common Core Math 8 is more rigorous than pre-Core Math 8 in most states (other than California) for which Math 8 does not equal Algebra I. Common Core Math 8 is more rigorous than non-Cali pre-Core Math 8 in that some (not all) Algebra I content has been added. But it's easy to get confused because this is an article in a Cali newspaper about a Cali district, and so Garelick thinks that she's trying to compare Common Core Math 8 to Cali pre-Core Math 8 (= Algebra I).

Garelick compares Common Core Math to his favorite Dolciani text from 50 years ago:

I supplement freely with a pre-algebra book by Dolciani written in the 70’s and other materials. The emphasis on ratio and proportion in 7th and 8th grades is rather drawn out and can be done more concisely, rather than harping on what a direct variation and proportional relationship is. Traditional Algebra 1 courses present direct variation in a much more understandable way, rather than the “beating around the bush” technique that defines such relationships as straight line functions that go through the origin, and whose slope equals the “constant of variation/proportionality”.

Recall that I found a copy of a 1970 Dolciani text at a library book sale. But I could find no mention of direct variation in the text. (My text is called "Course 2" -- I suspect that Garelick actually uses "Course 3" in his classroom.)

I look at my other 1960's-era text, Moise's Geometry, and I notice this in the preface:

"In recent years, there has been extensive discussion about the content of the geometry course ordinarily taught in the tenth grade."

So to Moise, Geometry is a sophomore course, yet to the traditionalists who prefer texts from his era, Geometry is a freshman course. The idea that eighth graders should be in Algebra I or seniors in Calculus is a fairly recent one. It never occurred to Moise, Dolciani, and other textbook writers of the 1960's and '70's that Calculus should be taught in high school.

But the real goal of San Francisco’s elimination of algebra in 8th grade is to close the achievement gap as evidenced by the last paragraph in the article.

Traditionalists don't wish to close the achievement gap -- instead, they favor tracking, which is the exact opposite. If most of the students in Garelick's eighth grade Algebra I class or the AP Calculus class are members of privileged groups (with "privilege" as defined by Eugenia Cheng), then so be it.

SteveH returns to comment in this thread:

SteveH:
Astounding. Why not eliminate the leveled groups in K-6 they use for differentiated instruction? It all doesn’t make any sense to the most casual observer. CCSS has a slope that leads to no remediation in college algebra – they say this! – but Jo Boaler, etal. claim that it’s normal to magically change that slope in high school to get to calculus, a level difficult even for those who get algebra in 8th grade.

Recall that Boaler wrote the preface to the Number Talks book, so I assume that she supports the methods used in that book. One of its authors, Cathy Humphreys, actually does mention Calculus in her book (in the chapter on fractions). Let me provide the full context:

"One day, as Cathy was working with sixth graders to help them find different ways to compare fractions, the class was unusually passive -- and almost sullen. Finally she stopped and asked what was wrong. After a minute or so, Anthony spoke up, and, even though it was some years ago, his words are still etched in her mind: 'Mrs. Humphreys, we had fractions in third grade and fourth grade and fifth grade. We didn't get them then, and we don't get them now -- and we don't want to do them anymore!' Not being able to 'get' fractions made Anthony feel unsuccessful -- and who wants to work on things that make us feel like that?

"But for success in high school, there is no avoiding fractions. Students who are successfully learning complex concepts in algebra, trigonometry, and calculus can become confounded by a fraction in the middle of an equation."

Notice that Anthony, the sixth grader, doesn't want to do fractions. He'd much rather leave a problem set blank than answer Question #1 if it contains a fraction.

We know what solution Cathy Humphreys and Jo Boaler would recommend -- start doing Number Talks on fractions. But to Garelick, SteveH, and other traditionalists, anything other than traditional math would block Anthony from getting to AP Calc.

OK, then, so I'd like to see what the traditionalists would do with a student like Anthony. He makes it clear that he doesn't want to do fractions, so assume that he'd refuse to answer Question #1 on a traditional p-set with fractions. Go on, traditionalists, show us your magic!

Lesson 5-2 of the U of Chicago text is called "Types of Quadrilaterals." In the modern Third Edition of the text, quadrilaterals appear in Lesson 6-4.

This is what I wrote two years ago about today's lesson:

Lesson 5-2 of the U of Chicago text covers the various types of quadrilaterals. There are no theorems in this section, but just definitions. The concept of definition is important to the study of geometry, and in no lesson so far are definitions more prominent than in this lesson.

The lesson begins by defining parallelogram , rhombus , rectangle , and square . There's nothing wrong with any of those definitions. But then we reach a controversial definition -- that of trapezoid :

Definition:
A quadrilateral is a trapezoid if and only if it has at least one pair of parallel sides.
(emphasis mine)

Just as with the definition of parallel back in Lesson 1-7, we have two extra words that distinguish this from a traditional definition of trapezoid -- "at least." In other textbooks, no parallelogram is a trapezoid, but in the U of Chicago text, every parallelogram is a trapezoid!

To understand what's going on here, let's go back to the first geometer who defined some of the terms in the quadrilateral hierarchy -- of course, I'm talking about Euclid:

Of quadrilateral figures, a square is that which is both equilateral and right-angled an oblong that which is right-angled but not equilateral a rhombus that which is equilateral but not right-angled and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

Of course, the modern term for "oblong" is rectangle , and a "rhomboid" is now a parallelogram . The word "trapezia" is actually plural of "trapezium." In British English, a "trapezium" is what we Americans would call a trapezoid , but to Euclid, any quadrilateral that is not a parallelogram (or below on the quadrilateral hierarchy) is a "trapezium." But the important part here is that to Euclid, a square, for example, is neither a rectangle (oblong) nor a rhombus. He makes sure to say that a rectangle (oblong) is "not equilateral," and that a rhombus is "not right-angled." And of course, neither a rectangle nor a rhombus is a parallelogram (rhomboid).

These are called "exclusive" definitions. For Euclid, there was no quadrilateral hierarchy -- each class of quadrilaterals was disjoint from the others. But since the days of Euclid, more and more geometry texts have slowly added more "inclusive" definitions.

One of the first inclusive definitions I've seen was the definition of rectangle . It was mentioned on an episode of Square One TV, when a Pacman parody character named Mathman was supposed to eat rectangles, and then ate a square because "every square's a rectangle." I would provide a YouTube link, but I haven't found the link in years and it doesn't come up in a search. (I even remember someone posting in the comments that just as for me, his first encounter of the inclusive definition of rectangle was through watching that clip when it first aired so many years ago!)

But many of my family members were also teachers, and one relative gave me an old textbook that still mentioned some exclusive definitions. In particular, it declared that a square isn't a rhombus. A little later, my fifth grade teacher then taught the inclusive definition of rhombus . I then blurted out that a square isn't a rhombus, then actually brought the old text to school to prove it! She replied, "Wow!" but then, if I recalled correctly, told me that this definition was old, and that by the new definition, a square is a rhombus. And so all modern texts classify the square as both a rectangle and a rhombus, and that all of these are considered parallelograms.

So we see that there is a tendency for definitions to grow more inclusive as time goes on. (We see this happening in politics as well -- for example, the definition of marriage . But I digress.) And so we see the next natural step is for the parallelogram to be considered a trapezoid.

One of the first advocates I saw for an inclusive definition of trapezoid is the famous Princeton mathematician John H. Conway. He is best known for inventing the mathematical Game of Life, which has its own website:

But Conway also specializes in other fields of mathematics, such as geometry and group theory (which is, in some ways, the study of symmetry). Twelve years ago, he posted the following information about why he prefers inclusive definitions:

The preference for exclusive definitions arises, I think, from
what I call "the descriptive use". Of course, one wouldn't DESCRIBE
a square table as "rectangular", since that would wantonly use
a longer term to convey less information. So in descriptive uses,
there's a natural presumption that a table called "rectangular"
won't in fact be square - in other words, a natural presumption
that the terms will be used exclusively.

But the descriptive use is unimportant to geometry, where the
really important thing is the truth of theorems. This means that we
should use a term "A" to include "B" if all the identities that
hold for all "A"s will also hold for all "B"s (in the way that
the trapezoid area theorem holds for all parallelograms, for
instance).

You might worry about the consistency of switching to the
inclusive use while other people continue with the exclusive one.
But there can be no consistency with people who are inconsistent!
I've seen many geometry books that MAKE the exclusive definitions,
but none that manage to USE them consistently for more than a few
theorems.

Indeed, Conway advocated taking it one step forward and actually abolishing the trapezoid and having only the isosceles trapezoid in the hierarchy! After all, there's not much one can say about a trapezoid that's not isosceles -- just look ahead to Lesson 5-5. There's only one theorem listed there about general trapezoids -- the Trapezoid Angle Theorem, and that's really just the Same-Side Interior Angle Consequence Theorem that can be proved without reference to trapezoids at all. All the other theorems in the lesson refer to isosceles trapezoids. In particular, the symmetry theorems in the lesson refer to isosceles trapezoids. (Recall that Conway's specializes in group theory -- which as I wrote above is the study of symmetry .) I suspect that the only reason that we have general trapezoids is that they are the simplest quadrilateral for which an area formula can be given.

This is now another digression from Common Core Geometry, so I'll just provide another link. Notice that here, Conway also proposes a hexagon hierarchy based on symmetry. There's also a pentagon hierarchy, but there are only three types of pentagons -- general, line-symmetric, and regular -- just as there are for triangles. It's easier to make figures with an even number of sides symmetric.

Another advocate of inclusive definitions is Mr. Chase, a Maryland high school math teacher. (And no, in yesterday's post and today's I'm referring to two different Maryland teachers.) I see that he is so passionate about the inclusive definition of trapezoid that he devoted three whole blog posts to why he hates the exclusive definition of trapezoid :

One reason Chase states for using inclusive definitions is that it simplifies proofs:

When proving that a quadrilateral is a trapezoid, one can stop after proving just two sides are parallel . But with the exclusive definition, in order to prove that a quadrilateral is a trapezoid, you would have to prove two sides are parallel AND the other two sides are not parallel.

Regarding some of the others to whom I refer regularly, Dr. Wu uses the inclusive definition:

"A quadrilateral with at least one pair of opposite sides that are parallel is called a trapezoid. A trapezoid with two pairs of parallel opposite sides is called a parallelogram."

while Dr. Mason uses the exclusive definition:

"A trapezoid is by definition a quadrilateral with precisely one pair of parallel sides."
(emphasis Dr. M's)

So which definition should I use for trapezoid ? Well, this is a Common Core blog, so the definition favored by Common Core should have priority over all other definitions. The following is a link to the information that will appear on the PARCC End-of-Year Assessment for geometry:

And right there in the column under "Clarifications," it reads:

i) A trapezoid is defined as “A quadrilateral with at least one pair of parallel sides.”

And that plainly settles it. The PARCC Common Core assessment uses the inclusive definition of trapezoid , and so it's my duty on a Common Core blog to use the Common Core definition. Of course, we notice that this is the definition given by PARCC -- but so far I've seen no information on what definition Smarter Balanced is using. It would be tragic if PARCC were to use one definition and Smarter Balanced the other. But as I can't say anything about Smarter Balanced, I will use the only definition that's known to be on a Common Core test, and that's the inclusive definition. The fact that this definition is already used by the U of Chicago is icing on the cake.

There is one problem with the inclusive definition of trapezoid , and that's when we try to define isosceles trapezoid . The word isosceles suggests that, just as in an isosceles triangle, an isosceles trapezoid has two equal sides -- the sides adjacent to the (parallel) bases. But in a parallelogram, where either pair of opposite sides can be considered the bases -- the sides adjacent to these bases are also equal. This would make every parallelogram an isosceles trapezoid. But this isn't desirable -- an isosceles trapezoid has several properties that parallelograms in general lack. The diagonals of an isosceles trapezoid are equal, but those of a parallelogram in general aren't. But the diagonals of a rectangle are equal. So we'd like to consider rectangles, but not parallelograms in general, to be isosceles trapezoids.

[2018 update: According to the old Moise text I mentioned above, all parallelograms are isosceles trapezoids while no kite is a rhombus. Otherwise his definition of trapezoid matches U of Chicago's.]

This dilemma is mentioned in the comments at one of the Chase links. It's pointed out that there are two ways out of this mess -- we may either define isosceles trapezoid in terms of symmetry, as Conway does, or we can use the U of Chicago's definition:

"A trapezoid is isosceles if and only if it has a pair of base angles equal in measure."

Some don't like this definition, because it violates linguistic purity -- the word isosceles comes from Greek, and it means "equal legs," not "equal angles." But as it turns out, it's a small price to pay to make the quadrilateral hierarchy and other theorems work out. And besides, any geometer who calls a nine-sided polygon a nonagon should just shut up about linguistic purity!

Note: Mr. Chase has made only one post in 2018. It's all about using pure Geometry to prove identities in Trig.


Geometry, Common Core Style

This is what Theoni Pappas writes on page 303 of her Magic of Mathematics:

"Using your visualization skills and folding techniques, determine a way to make one straight cut to separate the checkerboard into 2 * 2 squares, such as this one."

This is the second page of the subsection "Checkerboard Mania." Once again, the first page of this section was blocked by the weekend.

But we don't need to see the first page -- nor, for that matter, the pictures on this page -- in order to understand the problem. The checkerboard is a standard 8 * 8, and we are being asked to fold it so that a single cut will divide it into sixteen 2 * 2 squares. Pappas describes it as "taking a checkerboard apart with one fell swoop."

As usual, I'll post the solution tomorrow. This problem isn't easy at all -- the folds and the cut to make is extremely clever.

Lesson 5-2 of the U of Chicago text is called "Types of Quadrilaterals." In the modern Third Edition of the text, quadrilaterals appear in Lesson 6-4.

This is what I wrote two years ago about today's lesson:

Lesson 5-2 of the U of Chicago text covers the various types of quadrilaterals. There are no theorems in this section, but just definitions. The concept of definition is important to the study of geometry, and in no lesson so far are definitions more prominent than in this lesson.

The lesson begins by defining parallelogram , rhombus , rectangle , and square . There's nothing wrong with any of those definitions. But then we reach a controversial definition -- that of trapezoid :

Definition:
A quadrilateral is a trapezoid if and only if it has at least one pair of parallel sides.
(emphasis mine)

Just as with the definition of parallel back in Lesson 1-7, we have two extra words that distinguish this from a traditional definition of trapezoid -- "at least." In other textbooks, no parallelogram is a trapezoid, but in the U of Chicago text, every parallelogram is a trapezoid!

To understand what's going on here, let's go back to the first geometer who defined some of the terms in the quadrilateral hierarchy -- of course, I'm talking about Euclid:

Of quadrilateral figures, a square is that which is both equilateral and right-angled an oblong that which is right-angled but not equilateral a rhombus that which is equilateral but not right-angled and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

Of course, the modern term for "oblong" is rectangle , and a "rhomboid" is now a parallelogram . The word "trapezia" is actually plural of "trapezium." In British English, a "trapezium" is what we Americans would call a trapezoid , but to Euclid, any quadrilateral that is not a parallelogram (or below on the quadrilateral hierarchy) is a "trapezium." But the important part here is that to Euclid, a square, for example, is neither a rectangle (oblong) nor a rhombus. He makes sure to say that a rectangle (oblong) is "not equilateral," and that a rhombus is "not right-angled." And of course, neither a rectangle nor a rhombus is a parallelogram (rhomboid).

These are called "exclusive" definitions. For Euclid, there was no quadrilateral hierarchy -- each class of quadrilaterals was disjoint from the others. But since the days of Euclid, more and more geometry texts have slowly added more "inclusive" definitions.

One of the first inclusive definitions I've seen was the definition of rectangle . It was mentioned on an episode of Square One TV, when a Pacman parody character named Mathman was supposed to eat rectangles, and then ate a square because "every square's a rectangle." I would provide a YouTube link, but I haven't found the link in years and it doesn't come up in a search. (I even remember someone posting in the comments that just as for me, his first encounter of the inclusive definition of rectangle was through watching that clip when it first aired so many years ago!)

But many of my family members were also teachers, and one relative gave me an old textbook that still mentioned some exclusive definitions. In particular, it declared that a square isn't a rhombus. A little later, my fifth grade teacher then taught the inclusive definition of rhombus . I then blurted out that a square isn't a rhombus, then actually brought the old text to school to prove it! She replied, "Wow!" but then, if I recalled correctly, told me that this definition was old, and that by the new definition, a square is a rhombus. And so all modern texts classify the square as both a rectangle and a rhombus, and that all of these are considered parallelograms.

So we see that there is a tendency for definitions to grow more inclusive as time goes on. (We see this happening in politics as well -- for example, the definition of marriage . But I digress.) And so we see the next natural step is for the parallelogram to be considered a trapezoid.

One of the first advocates I saw for an inclusive definition of trapezoid is the famous Princeton mathematician John H. Conway. He is best known for inventing the mathematical Game of Life, which has its own website:

But Conway also specializes in other fields of mathematics, such as geometry and group theory (which is, in some ways, the study of symmetry). Twelve years ago, he posted the following information about why he prefers inclusive definitions:

The preference for exclusive definitions arises, I think, from
what I call "the descriptive use". Of course, one wouldn't DESCRIBE
a square table as "rectangular", since that would wantonly use
a longer term to convey less information. So in descriptive uses,
there's a natural presumption that a table called "rectangular"
won't in fact be square - in other words, a natural presumption
that the terms will be used exclusively.

But the descriptive use is unimportant to geometry, where the
really important thing is the truth of theorems. This means that we
should use a term "A" to include "B" if all the identities that
hold for all "A"s will also hold for all "B"s (in the way that
the trapezoid area theorem holds for all parallelograms, for
instance).

You might worry about the consistency of switching to the
inclusive use while other people continue with the exclusive one.
But there can be no consistency with people who are inconsistent!
I've seen many geometry books that MAKE the exclusive definitions,
but none that manage to USE them consistently for more than a few
theorems.

Indeed, Conway advocated taking it one step forward and actually abolishing the trapezoid and having only the isosceles trapezoid in the hierarchy! After all, there's not much one can say about a trapezoid that's not isosceles -- just look ahead to Lesson 5-5. There's only one theorem listed there about general trapezoids -- the Trapezoid Angle Theorem, and that's really just the Same-Side Interior Angle Consequence Theorem that can be proved without reference to trapezoids at all. All the other theorems in the lesson refer to isosceles trapezoids. In particular, the symmetry theorems in the lesson refer to isosceles trapezoids. (Recall that Conway's specializes in group theory -- which as I wrote above is the study of symmetry .) I suspect that the only reason that we have general trapezoids is that they are the simplest quadrilateral for which an area formula can be given.

This is now another digression from Common Core Geometry, so I'll just provide another link. Notice that here, Conway also proposes a hexagon hierarchy based on symmetry. There's also a pentagon hierarchy, but there are only three types of pentagons -- general, line-symmetric, and regular -- just as there are for triangles. It's easier to make figures with an even number of sides symmetric.

Another advocate of inclusive definitions is Mr. Chase, a Maryland high school math teacher. (And no, in yesterday's post and today's I'm referring to two differentMaryland teachers.) I see that he is so passionate about the inclusive definition of trapezoid that he devoted three whole blog posts to why he hates the exclusive definition of trapezoid :

One reason Chase states for using inclusive definitions is that it simplifies proofs:

When proving that a quadrilateral is a trapezoid, one can stop after proving just two sides are parallel . But with the exclusive definition, in order to prove that a quadrilateral is a trapezoid, you would have to prove two sides are parallel AND the other two sides are not parallel.

Regarding some of the others to whom I refer regularly, Dr. Wu uses the inclusive definition:

"A quadrilateral with at least one pair of opposite sides that are parallel is called a trapezoid. A trapezoid with two pairs of parallel opposite sides is called a parallelogram."

while Dr. Mason uses the exclusive definition:

"A trapezoid is by definition a quadrilateral with precisely one pair of parallel sides."
(emphasis Dr. M's)

So which definition should I use for trapezoid ? Well, this is a Common Core blog, so the definition favored by Common Core should have priority over all other definitions. The following is a link to the information that will appear on the PARCC End-of-Year Assessment for geometry:

And right there in the column under "Clarifications," it reads:

i) A trapezoid is defined as “A quadrilateral with at least one pair of parallel sides.”

And that plainly settles it. The PARCC Common Core assessment uses the inclusive definition of trapezoid , and so it's my duty on a Common Core blog to use the Common Core definition. Of course, we notice that this is the definition given by PARCC -- but so far I've seen no information on what definition Smarter Balanced is using. It would be tragic if PARCC were to use one definition and Smarter Balanced the other. But as I can't say anything about Smarter Balanced, I will use the only definition that's known to be on a Common Core test, and that's the inclusive definition. The fact that this definition is already used by the U of Chicago is icing on the cake.

There is one problem with the inclusive definition of trapezoid , and that's when we try to define isosceles trapezoid . The word isosceles suggests that, just as in an isosceles triangle, an isosceles trapezoid has two equal sides -- the sides adjacent to the (parallel) bases. But in a parallelogram, where either pair of opposite sides can be considered the bases -- the sides adjacent to these bases are also equal. This would make every parallelogram an isosceles trapezoid. But this isn't desirable -- an isosceles trapezoid has several properties that parallelograms in general lack. The diagonals of an isosceles trapezoid are equal, but those of a parallelogram in general aren't. But the diagonals of a rectangle are equal. So we'd like to consider rectangles, but not parallelograms in general, to be isosceles trapezoids.

This dilemma is mentioned in the comments at one of the Chase links. It's pointed out that there are two ways out of this mess -- we may either define isosceles trapezoid in terms of symmetry, as Conway does, or we can use the U of Chicago's definition:

"A trapezoid is isosceles if and only if it has a pair of base angles equal in measure."

Some don't like this definition, because it violates linguistic purity -- the word isosceles comes from Greek, and it means "equal legs," not "equal angles." But as it turns out, it's a small price to pay to make the quadrilateral hierarchy and other theorems work out. And besides, any geometer who calls a nine-sided polygon a nonagon should just shut up about linguistic purity!

By the way, Mr. Chase has returned to posting in 2017, at least in January through April. His most recent post is about the biennial National Math Festival, held in Washington, DC.


9.7: Use Properties of Rectangles, Triangles, and Trapezoids (Part 2) - Mathematics

I introduce Properties of Trapezoids and the Mid-segment Theorem
EXAMPLES AT 3:56 7:54 13:20

Course Index

  1. Points Lines & Planes in Geometry
  2. More Postulates & Theorems Points, Lines, & Planes
  3. Segment, Ray, Distance on a Number Line
  4. Introduction to Angles
  5. Special Angle Pairs
  6. Basic Constructions Segment & Angle
  7. Angle and Segment Bisector Constructions
  8. Distance Formula & Pythagorean Theorem
  9. Midpoint Formula
  10. Perimeter of a Plane Region
  11. Area of a Plane Region
  12. Deductive Reasoning If Then Statements
  13. Intro to 2 Column Geometry Proofs
  14. Congruent Angles
  15. Parallel Lines & Skew Lines, Angles formed by a Transversal
  16. Parallel Lines & Transversal Angles
  17. Proving Lines Parallel
  18. Parallel & Perpendicular Lines Proof
  19. Angles of Triangles and Parallel Lines
  20. Graphing Lines in Slope-Intercept form y=mx+b
  21. Equations of Lines and Graphing
  22. Rate of Change Slope & Point Slope Equation of Lines
  23. Equations of Parallel and Perpendicular Lines
  24. Congruent Polygons & Third Angle Theorem
  25. Congruent Triangles SSS SAS
  26. Congruent Triangles ASA AAS
  27. Non Congruence Theorems AAA SSA
  28. CPCTC Corresponding Parts of Congruent Triangles are Congruent
  29. Isosceles & Equilateral Triangle Properties
  30. Equilateral Triangle / Equiangular Triangle
  31. Congruent Right Triangles HL Hypotenuse Leg Theorem
  32. Triangle Midsegment Theorem
  33. Proving Midsegment Theorem
  34. Perpendicular Bisector Theorem
  35. Angle Bisector Theorem
  36. Triangle Perpendicular Bisectors & Circumcenter
  37. Triangle Angle Bisectors & Incenter
  38. Medians of Triangles & Centroid
  39. Altitudes of Triangles and Orthocenter
  40. Triangle Inequality Theorem
  41. Hinge Theorem Inequalities 2 Triangles
  42. Polygon Angle Sum Theorems
  43. Properties of Parallelograms
  44. Proving Quadrilaterals are Parallelograms
  45. Rhombuses, Rectangles, and Squares
  46. More Examples Rhombus & Rectangle
  47. Properties of Trapezoids & Mid Segment Theorem
  48. Properties of Kites
  49. Polygons in Coordinate Plane
  50. Ratios & Proportions
  51. Proportions and Similar Polygons / Similar Figures
  52. Proving Triangles are Similar AA SAS SSS
  53. Similarity in Right Triangles
  54. Side Splitter Theorem Proportions in Triangles
  55. Proportions in Triangles Angle Bisector
  56. Pythagorean Theorem
  57. Converse Pythagorean Theorem & Triples
  58. Special Right Triangles 45-45-90 30-60-90
  59. Right Triangle Trigonometry Part 1: Finding Missing Sides
  60. Right Triangle Trigonometry Part 2: Solving for Acute Angles
  61. Angle of Elevation and Depression Right Triangle Trig
  62. Area Parallelograms and Triangles
  63. Area Trapezoid Rhombus Kite
  64. Area of Regular Polygon Introduction with Hexagon Examples
  65. Area Regular Polygons with Trigonometry
  66. Perimeter and Area Ratios of Similar Figures
  67. Circle Intro and Arc Length
  68. Sector and Segment Area in Circles
  69. Segment Area in Circles Quicker Methods
  70. Geometric Probabilities Length
  71. Geometric Probabilities Area
  72. Surface Area of a Right Prism
  73. Surface Area Of a Cylinder
  74. Surface Area of a Regular Pyramid Non-Trig Examples
  75. Surface Area of a Pyramid with Trigonometry
  76. Surface Area of a Cone
  77. Volume of a Prism & Volume of a Cylinder
  78. Volume of a Pyramid 3 Examples
  79. Volume of a Cone 3 Examples
  80. Surface Area of a Sphere & Volume of a Sphere
  81. Tangent Lines to Circles
  82. Given a Tangent Line & Circle Find the Point of Tangency
  83. Chords Arcs and Diameters in Circle
  84. Inscribed Angles in Circles and Tangent Lines
  85. Angles in Circles Chords Secants Tangents and Arcs
  86. Segment Lengths in Circles with Chords, Secants, and Tangents

Course Description

In this series, the very helpful and fun math teacher Mr. Tarrou teaches students an entire course on geometry from start to finish. His videos are friendly, easy to understand, entertaining, and very well organized, all thanks to Mr. Tarrou great dedication to teaching and mathusiasm


This set of geometry challenges focuses on creating a variety of polygons as students problem solve and think as they learn to code using block coding software. Student will need to use their knowledge of the attributes of polygons and mathematical principals of geometry to accomplish the given challenges. The challenges start out fairly simple and move to more complex situations in which students can explore at their own pace or work as a team. Computer Science standards are seamlessly intertwined with the math standards while providing “Step it up!” and “Jump it up!” opportunities to increase rigor.

In this lesson students will use standards-based quadrilaterals and triangles to design a roller coaster tower. Students will use the Engineering Design Process to work through the processes in this lesson.

In this lesson, students will practice classifying and naming quadrilaterals. Students will take notes and use the notes to complete a quadrilaterals Venn diagram. Students will be asked to name and classify quadrilaterals using all applicable names.

"Where in the Venn are the Quadrilaterals?" is an activity that helps the student to develop a better understanding of classifying two-dimensional figures in a hierarchy based on properties.

During this activity, students will read a book about the Brooklyn Bridge. After whole class discussion, children will explore different types of bridges and data, in order to decipher which bridge is the strongest. The students will work collaboratively in groups with assigned student roles. Students will utilized Higher Order thinking to create a solution. The culminating activity is a presentation of solution to whole class.

Students will construct several simple polyhedra, then count the number of faces, edges, and vertices. These data should suggest Euler's formula.

The student will be engaged in a paper plane making activity while discovering the attributes of different triangles. The students will learned the similarities and differences of the following triangles: scalene, isosceles, equilateral, right, obtuse, and acute.


Properties of Rectangles, Parallelograms and Trapezoids

Videos, games, activities and worksheets to help ACT students review properties of rectangles, parallelograms and trapezoids. Properties of Special Parallelograms - rhombus, rectangle, square :
Squares and rectangles are special types of parallelograms with special properties. A square is a type of equiangular parallelogram and square properties include congruent diagonals and diagonals that bisect each other. A rectangle is a type of regular quadrilateral. Rectangle properties include (1) diagonals that are congruent, (2) perpendicular diagonals that bisect each other and (3) diagonals that bisect each of the angles.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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Watch the video: τριγωνομετρικες εξισωσειςπαρ13 (November 2021).