5.13: Trapezoids
- Page ID
- 4997
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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](https://k12.libretexts.org/@api/deki/files/1819/f-d_85da86b74d3e7cff769a38ed51bf8ea9dd6bfb06b142c8d8100be848%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
An isosceles trapezoid is a trapezoid where the non-parallel sides are congruent.
![f-d_6692d09bbbeeefe05e18e626314c63b9c7c8f412a3290ee10cae837d+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/1820/f-d_6692d09bbbeeefe05e18e626314c63b9c7c8f412a3290ee10cae837d%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
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](https://k12.libretexts.org/@api/deki/files/1821/f-d_90ad2cc14decdd426fb2b247e7621150dcf950854e8cfa0d4f11b876%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
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](https://k12.libretexts.org/@api/deki/files/1822/f-d_e7c3fc7c597d0f75dc3e68eb9a47affaa3da6a09a4805f244fffb011%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
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](https://k12.libretexts.org/@api/deki/files/1823/f-d_e9b587e86482f04cb836938279acfe16bc94e5f454261bd57fe7026e%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
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](https://k12.libretexts.org/@api/deki/files/1824/f-d_c00ace22483d21997564bddc94c062df030a05d979ec7c699c55246c%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
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](https://k12.libretexts.org/@api/deki/files/1825/f-d_bfb1a31bd2f7b79971801bac7c3c3dbe4b00a51e4a6caf107c9f032c%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
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](https://k12.libretexts.org/@api/deki/files/1826/f-d_053ad68c2d327c0fe2250dae3f8075e1a661132b221a972e194ec02d%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
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.
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Figure \(\PageIndex{10}\)
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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.
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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).
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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