Trapezoids
Bill Zahner Lori Jordan
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AUTHORS Bill Zahner Lori Jordan
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C HAPTER
Chapter 1. Trapezoids
1
Trapezoids
Here you’ll learn the properties of trapezoids and how to apply them. What if you were told that the polygon ABCD is an isoceles trapezoid and that one of its base angles measures 38◦ ? What can you conclude about its other angles? After completing this Concept, you’ll be able to find the value of a trapezoid’s unknown angles and sides given your knowledge of the properties of trapezoids.
Guidance
A trapezoid is a quadrilateral with exactly one pair of parallel sides. Examples look like:
An isosceles trapezoid is a trapezoid where the non-parallel sides are congruent. The third trapezoid above is an example of an isosceles trapezoid. Think of it as an isosceles triangle with the top cut off. Isosceles trapezoids also have parts that are labeled much like an isosceles triangle. Both parallel sides are called bases.
Recall that in an isosceles triangle, the two base angles are congruent. This property holds true for isosceles trapezoids. Theorem: The base angles of an isosceles trapezoid are congruent. The converse is also true: If a trapezoid has congruent base angles, then it is an isosceles trapezoid. Next, we will investigate the diagonals of an isosceles trapezoid. Recall, that the diagonals of a rectangle are congruent AND they bisect each other. The diagonals of an isosceles trapezoid are also congruent, but they do NOT bisect each other. Isosceles Trapezoid Diagonals Theorem: The diagonals of an isosceles trapezoid are congruent. The midsegment (of a trapezoid) is a line segment that connects the midpoints of the non-parallel sides. There is only one midsegment in a trapezoid. It will be parallel to the bases because it is located halfway between them. Similar to the midsegment in a triangle, where it is half the length of the side it is parallel to, the midsegment of a trapezoid also has a link to the bases. 1
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Investigation: Midsegment Property
Tools Needed: graph paper, pencil, ruler 1. Draw a trapezoid on your graph paper with vertices A(−1, 5), B(2, 5), C(6, 1) and D(−3, 1). Notice this is NOT an isosceles trapezoid.
2. Find the midpoint of the non-parallel sides either by using slopes or the midpoint formula. Label them E and F. Connect the midpoints to create the midsegment. 3. Find the lengths of AB, EF, and CD. Can you write a formula to find the midsegment? Midsegment Theorem: The length of the midsegment of a trapezoid is the average of the lengths of the bases, or EF = AB+CD . 2
2
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Chapter 1. Trapezoids
Example A
Look at trapezoid T RAP below. What is m6 A?
Find x. All figures are trapezoids with the midsegment.
Guided Practice
T RAP an isosceles trapezoid.
Find: 1. 2. 3. 4.
m6 m6 m6 m6
T PA PT R PZA ZRA
3
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Practice
1. Can the parallel sides of a trapezoid be congruent? Why or why not? For questions 2-7, find the length of the midsegment or missing side.
2.
3.
4.
5.
6.
7. Find the value of the missing variable(s).
8. 4
