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# Trapezoids

## Quadrilaterals with exactly one pair of parallel sides.

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Trapezoids

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

### Trapezoids

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.

#### 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)\begin{align*}A(-1, 5), \ B( 2, 5), \ C(6, 1)\end{align*} and D(3,1)\begin{align*}D(-3, 1)\end{align*}. 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\begin{align*}E\end{align*} and F\begin{align*}F\end{align*}. Connect the midpoints to create the midsegment.
3. Find the lengths of AB, EF\begin{align*}AB, \ EF\end{align*}, and CD\begin{align*}CD\end{align*}. 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+CD2\begin{align*}EF=\frac{AB+CD}{2}\end{align*}.

#### Measuring Angles

Look at trapezoid TRAP\begin{align*}TRAP\end{align*} below. What is mA\begin{align*}m \angle A\end{align*}?

TRAP\begin{align*}TRAP\end{align*} is an isosceles trapezoid. So, mR=115\begin{align*}m \angle R = 115^\circ\end{align*}. To find mA\begin{align*}m \angle A\end{align*}, set up an equation.

115+115+mA+mP230+2mA2mAmA=360=360mA=mP=130=65\begin{align*}115^\circ + 115^\circ + m \angle A + m \angle P & = 360^\circ\\ 230^\circ + 2m \angle A & = 360^\circ \rightarrow m \angle A = m \angle P\\ 2m \angle A & = 130^\circ\\ m \angle A & = 65^\circ\end{align*}

Notice that mR+mA=115+65=180\begin{align*}m \angle R + m \angle A = 115^\circ + 65^\circ = 180^\circ\end{align*}. These angles will always be supplementary because of the Consecutive Interior Angles Theorem. Therefore, the two angles along the same leg (or non-parallel side) are always going to be supplementary. Only in isosceles trapezoids will opposite angles also be supplementary.

#### Writing a Two-Column Proof

Write a two-column proof.

Given: Trapezoid ZOID\begin{align*}ZOID\end{align*} and parallelogram ZOIM\begin{align*}ZOIM\end{align*}

DI\begin{align*}\angle D \cong \angle I\end{align*}

Prove: ZD¯¯¯¯¯¯¯¯OI¯¯¯¯¯¯\begin{align*}\overline{ZD} \cong \overline{OI}\end{align*}

Statement Reason
1. Trapezoid ZOID\begin{align*}ZOID\end{align*} and parallelogram ZOIM, DI\begin{align*}ZOIM, \ \angle D \cong \angle I\end{align*} Given
2. ZM¯¯¯¯¯¯¯¯¯OI¯¯¯¯¯¯\begin{align*}\overline{ZM} \cong \overline{OI}\end{align*} Opposite Sides Theorem
3. IZMD\begin{align*}\angle I \cong \angle ZMD\end{align*} Corresponding Angles Postulate
4. DZMD\begin{align*}\angle D \cong \angle ZMD\end{align*} Transitive PoC
5. ZM¯¯¯¯¯¯¯¯¯ZD¯¯¯¯¯¯¯¯\begin{align*}\overline{ZM} \cong \overline{ZD}\end{align*} Base Angles Converse
6. ZD¯¯¯¯¯¯¯¯OI¯¯¯¯¯¯\begin{align*}\overline{ZD} \cong \overline{OI}\end{align*} Transitive PoC

#### Solving for Unknown Values

Find x\begin{align*}x\end{align*}. All figures are trapezoids with the midsegment.

1. a) x\begin{align*}x\end{align*} is the average of 12 and 26. 12+262=382=19\begin{align*}\frac{12+26}{2}=\frac{38}{2}=19\end{align*}

b) 24 is the average of x\begin{align*}x\end{align*} and 35.

x+352x+35x=24=48=13\begin{align*}\frac{x+35}{2}&=24\\ x+35&=48\\ x&=13\end{align*}

c) 20 is the average of 5x15\begin{align*}5x-15\end{align*} and 2x8\begin{align*}2x-8\end{align*}.

5x15+2x827x237xx=20=40=63=9\begin{align*}\frac{5x-15+2x-8}{2}&=20\\ 7x-23& =40\\ 7x&=63\\ x&=9\end{align*}

#### Earlier Problem Revisited

Given an isosceles trapezoid with mB=38\begin{align*} m \angle B = 38^\circ \end{align*}, find the missing angles.

In an isosceles trapezoid, base angles are congruent.

BC\begin{align*} \angle B \cong \angle C\end{align*} and AD\begin{align*}\angle A \cong \angle D\end{align*}

mB=mC=3838+38+mA+mD=360mA=mD=142\begin{align*}m \angle B = m \angle C = 38^\circ\\ 38^\circ + 38^\circ + m \angle A + m \angle D = 360^\circ\\ m \angle A = m \angle D = 142^\circ\end{align*}

### Examples

TRAP\begin{align*}TRAP\end{align*} an isosceles trapezoid.

Find:

#### m∠TPA\begin{align*}m \angle TPA\end{align*}

TPZRAZ\begin{align*}\angle TPZ \cong \angle RAZ\end{align*} so mTPA=20+35=55\begin{align*}m\angle TPA=20^\circ + 35^\circ = 55^\circ\end{align*}

#### m∠PTR\begin{align*}m \angle PTR\end{align*}

TPA\begin{align*}\angle TPA\end{align*} is supplementary with PTR\begin{align*}\angle PTR\end{align*}, so mPTR=125\begin{align*}m\angle PTR=125^\circ\end{align*}

#### m∠PZA\begin{align*}m \angle PZA\end{align*}

By the Triangle Sum Theorem35+35+mPZA=180\begin{align*}35^\circ + 35^\circ + m\angle PZA=180^\circ\end{align*}, so mPZA=110\begin{align*}m\angle PZA=110^\circ\end{align*}

#### m∠ZRA\begin{align*}m \angle ZRA\end{align*}

Since mPZA=110\begin{align*}m\angle PZA = 110^\circ\end{align*}, mRZA=70\begin{align*}m\angle RZA=70^\circ\end{align*} because they form a linear pair. By the Triangle Sum Theorem, mZRA=90\begin{align*}m\angle ZRA=90^\circ\end{align*}

### Review

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.

Find the value of the missing variable(s).

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

1. A(3,2),B(1,3),C(3,1),D(4,2)\begin{align*}A(-3, 2), B(1, 3), C(3, -1), D(-4, -2)\end{align*}
2. A(3,3),B(2,2),C(6,6),D(7,1)\begin{align*}A(-3, 3), B(2, -2), C(-6, -6), D(-7, 1)\end{align*}
3. A(1,3),B(4,0),C(2,4),D(3,1)\begin{align*}A(1, 3), B(4, 0), C(2, -4), D(-3, 1)\end{align*}
4. A(1,3),B(3,1),C(2,4),D(3,1)\begin{align*}A(1, 3), B(3, 1), C(2, -4), D(-3, 1)\end{align*}
1. Write a two-column proof of the Isosceles Trapezoid Diagonals Theorem using congruent triangles.

Given: TRAP\begin{align*}TRAP\end{align*} is an isosceles trapezoid with TR¯¯¯¯¯¯¯ || AP¯¯¯¯¯¯¯¯\begin{align*}\overline{TR} \ || \ \overline{AP}\end{align*}.

Prove: TA¯¯¯¯¯¯¯RP¯¯¯¯¯¯¯¯\begin{align*}\overline{TA} \cong \overline{RP}\end{align*}

1. How are the opposite angles in an isosceles trapezoid related?.
2. List all the properties of a trapezoid.

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### Vocabulary Language: English

TermDefinition
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.