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5.13: Trapezoids

Last updated

Nov 28, 2020
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- 5.12: Estimation of Parallelogram Area in Scale Drawings
- 5.14: Area and Perimeter of Trapezoids

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Determine unknown angle measurements of quadrilaterals with exactly one pair of parallel sides.

A trapezoid is a quadrilateral with exactly one pair of parallel sides.

f-d_85da86b74d3e7cff769a38ed51bf8ea9dd6bfb06b142c8d8100be848+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{1}$

An isosceles trapezoid is a trapezoid where the non-parallel sides are congruent.

f-d_6692d09bbbeeefe05e18e626314c63b9c7c8f412a3290ee10cae837d+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{2}$

The base angles of an isosceles trapezoid are congruent. If $$ABCD$ $ is an isosceles trapezoid, then $\angle A\cong \angle B$ and $\angle C\cong \angle D$ .

f-d_90ad2cc14decdd426fb2b247e7621150dcf950854e8cfa0d4f11b876+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{3}$

The converse is also true. If a trapezoid has congruent base angles, then it is an isosceles trapezoid. The diagonals of an isosceles trapezoid are also congruent. The midsegment (of a trapezoid) is a line segment that connects the midpoints of the non-parallel sides:

f-d_e7c3fc7c597d0f75dc3e68eb9a47affaa3da6a09a4805f244fffb011+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{4}$

There is only one midsegment in a trapezoid. It will be parallel to the bases because it is located halfway between them.

Midsegment Theorem: The length of the midsegment of a trapezoid is the average of the lengths of the bases.

f-d_e9b587e86482f04cb836938279acfe16bc94e5f454261bd57fe7026e+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{5}$

If $\overline{EF}$ is the midsegment, then $EF=\dfrac{AB+CD}{2}$ .

What if you were told that the polygon $ABCD$ is an isosceles trapezoid and that one of its base angles measures $38^{\circ}$ ? What can you conclude about its other base angle?

For Examples 1 and 2, use the following information:

$$TRAP$ $ is an isosceles trapezoid.

f-d_c00ace22483d21997564bddc94c062df030a05d979ec7c699c55246c+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{6}$

Example $\PageIndex{1}$

Find $m\angle TPA$ .

Solution

$\angle TPZ\cong \angle RAZ$ so $m\angle TPA=20^{\circ} +35^{\circ} =55^{\circ}$ .

Example $\PageIndex{2}$

Find $m\angle ZRA$ .

Solution

Since $m\angle PZA=110^{\circ}$ , $m\angle RZA=70^{\circ}$ because they form a linear pair. By the Triangle Sum Theorem, $m\angle ZRA=90^{\circ}$ .

Example $\PageIndex{3}$

Look at trapezoid $TRAP$ below. What is $m\angle A$ ?

f-d_bfb1a31bd2f7b79971801bac7c3c3dbe4b00a51e4a6caf107c9f032c+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{7}$

Solution

$TRAP$ is an isosceles trapezoid. $m\angle R=115^{\circ}$ also.

To find \(m\angle , set up an equation.

$\begin{aligned} 115^{\circ}+115^{\circ}+m \angle A+m \angle P &=360^{\circ} \\ 230^{\circ}+2 m \angle A &=360^{\circ} \quad \rightarrow m \angle A=m \angle P \\ 2 m \angle A &=130^{\circ} \\ m \angle A &=65^{\circ} \end{aligned}$

Notice that $m\angle R+m\angle A=115^{\circ} +65^{\circ} =180^{\circ}$ . These angles will always be supplementary because of the Consecutive Interior Angles Theorem.

Example $\PageIndex{4}$

Is $ZOID$ an isosceles trapezoid? How do you know?

f-d_053ad68c2d327c0fe2250dae3f8075e1a661132b221a972e194ec02d+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{8}$

Solution

$40^{\circ} \neq 35^{\circ}$ , $ZOID$ is not an isosceles trapezoid.

Example $\PageIndex{5}$

Find $x$ . All figures are trapezoids with the midsegment marked as indicated.

Figure $\PageIndex{10}$

Figure $\PageIndex{10}$
Figure $\PageIndex{11}$

Solution

$x$ is the average of 12 and 26. $\dfrac{12+26}{2}=\dfrac{38}{2}=19$
24 is the average of $x$ and 35.
$\begin{aligned} \dfrac{x+35}{2}&=24 \\ x+35&=48 \\ x&=13 \end{aligned}$

20 is the average of $5x−15$ and $2x−8$ .
$\begin{aligned} \dfrac{5x−15+2x−8}{2}&=20 \\ 7x−23&=40 \\ 7x&=63 \\ x&=9 \end{aligned}$

Review

1. Can the parallel sides of a trapezoid be congruent? Why or why not?

For questions 2-8, find the length of the midsegment or missing side.

Figure $\PageIndex{12}$
Figure $\PageIndex{13}$
Figure $\PageIndex{14}$
Figure $\PageIndex{15}$
Figure $\PageIndex{16}$
Figure $\PageIndex{17}$

Find the value of the missing variable(s).

Figure $\PageIndex{18}$

Find the lengths of the diagonals of the trapezoids below to determine if it is isosceles.

A(−3,2), B(1,3), C(3,−1), D(−4,−2)
A(−3,3), B(2,−2), C(−6,−6), D(−7,1)

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.6.

Vocabulary

Term	Definition
isosceles trapezoid	An *isosceles trapezoid* is a trapezoid where the non-parallel sides are congruent.
midsegment (of a trapezoid)	A line segment that connects the midpoints of the non-parallel sides.
trapezoid	A quadrilateral with exactly one pair of parallel sides.
Diagonal	A diagonal is a line segment in a polygon that connects nonconsecutive vertices
midsegment	A midsegment connects the midpoints of two sides of a triangle or the non-parallel sides of a trapezoid.

Additional Resources

Interactive Element

Video: Trapezoids Examples - Basic

Activities: Trapezoids Discussion Questions

Study Aids: Trapezoids and Kites Study Guide

Practice: Trapezoids

Real World: Trapezoids in Timbuktu

Review

Review (Answers)

Vocabulary

Additional Resources

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