Geometry (A): READ: Trapezoid Reading

Trapezoids

Learning Objectives

Understand and prove that the base angles of isosceles trapezoids are congruent.
Understand and prove that if base angles in a trapezoid are congruent, it is an isosceles trapezoid.
Understand and prove that the diagonals in an isosceles trapezoid are congruent.
Understand and prove that if the diagonals in a trapezoid are congruent, the trapezoid is isosceles.
Identify the median of a trapezoid and use its properties.
Identify that consecutive angles are supplementary.

Introduction

Trapezoids are particularly unique figures among quadrilaterals. They have exactly one pair of parallel sides so unlike rhombi, squares, and rectangles, they are not parallelograms. There are special relationships in trapezoids, particularly in isosceles trapezoids. Remember that isosceles trapezoids have non-parallel sides that are of the same lengths. They also have symmetry along a line that passes perpendicularly through both bases.

Isosceles Trapezoid

Non-isosceles Trapezoid

Base Angles in Isosceles Trapezoids

Previously, you learned about the Base Angles Theorem. The theorem states that in an isosceles triangle, the two base angles (opposite the congruent sides) are congruent. The same property holds true for isosceles trapezoids. The two angles along the same base in an isosceles triangle will also be congruent. Thus, this creates two pairs of congruent angles—one pair along each base.

Theorem: The base angles of an isosceles trapezoid are congruent

Example 1

Examine trapezoid $ABCD$ below.

What is the measure of angle $ADC$ ?

This problem requires two steps to solve. You already know that base angles in an isosceles triangle will be congruent, but you need to find the relationship between adjacent angles as well. Imagine extending the parallel segments $\overline {BC}$ and $\overline{AD}$ on the trapezoid and the transversal $\overline {AB}$ . You’ll notice that the angle labeled $115^\circ$ is a consecutive interior angle with $\angle {BAD}$ .

Consecutive interior angles along two parallel lines will be supplementary. You can find $m \angle{BAD}$ by subtracting $115^\circ$ from $180^\circ$ .

$m\angle{BAD} + 115^\circ & = 180^\circ \\ m\angle{BAD} & = 65^\circ$

So, $\angle {BAD}$ measures $65^\circ$ . Since $\angle{BCD}$ is adjacent to the same base as $\angle{ADC}$ in an isosceles trapezoid, the two angles must be congruent. Therefore, $m \angle {ADC} = 65^\circ.$

Leg angles of a Trapezoid are Supplementary

Each trapezoid has exactly one pair of parallel sides and one pair of opposite sides that are not parallel. The non-parallel sides of the trapezoid are called the legs of a trapezoid. Angles that touch the same leg of a trapezoid are called leg angles are are supplementary.

Identify Isosceles Trapezoids with Base Angles

In the last lesson, you learned about biconditional statements and converse statements. You just learned that if a trapezoid is an isosceles trapezoid then base angles are congruent. The converse of this statement is also true. If a trapezoid has two congruent angles along the same base, then it is an isosceles trapezoid. You can use this fact to identify lengths in different trapezoids.

First, we prove that this converse is true.

Theorem: If two angles along one base of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid

Given: Trapezoid $ZOID$ with $\overline {ZD} \| \overline {OI}$ and $\angle {OZD} \cong \angle {ZDI}$
Prove: $\overline {ZO} \cong \overline {ID}$

This proof is very similar to the previous proof, and it also relies on isosceles triangle properties.

Statement	Reason
1. Trapezoid $ZOID$ has $\overline {ZO}$ $\\|$ $\overline {OI}$ and $\angle {OZD} \cong \angle {ZDI}$	1. Given
2. Construct $\overline {OA}$ $\\|$ $\overline {ID}$	2. Parallel Postulate
3. $\angle {ZAO} \cong \angle {ADI}$	3. Corresponding Angles Postulate
4. $AOID$ is a parallelogram	4. Definition of a parallelogram
5. $\overline {AO} \cong \overline {ID}$	5. Opposite sides of a parallelogram are $\cong$
Trapezoid $ZOID$ with auxiliary lines
6. $\angle {OZA} \cong \angle {OAZ}$	6. Transitive Property
7. $\triangle {OZA}$ is isosceles	7. Definition of isosceles triangle
8. $\overline {OZ} \cong \overline {OA}$	8. Converse of the Base Angles Theorem
9. $\overline {OZ} \cong \overline {ID}$	9. Transitive Property $\blacklozenge$

Example 2

What is the length of $MN$ in the trapezoid below?

Notice that in trapezoid $LMNO$ , two base angles are marked as congruent. So, the trapezoid is isosceles. That means that the two non-parallel sides have the same length. Since you are looking for the length of $\overline {MN}$ , it will be congruent to $\overline {LO}$ . So, $MN = 3\;\mathrm{feet}$ .

Diagonals in Isosceles Trapezoids

The angles in isosceles trapezoids are important to study. The diagonals, however, are also important. The diagonals in an isosceles trapezoid will not necessarily be perpendicular as in rhombi and squares. They are, however, congruent. Any time you find a trapezoid that is isosceles, the two diagonals will be congruent.

Theorem: The diagonals of an isosceles trapezoid are congruent

Example 3

Review the two-column proof below.

Given: $WXYZ$ is a trapezoid and $\overline {WZ} \cong \overline {XY}$
Prove: $\overline {WY} \cong \overline {XZ}$

Statement	Reason
1. $\overline {WZ} \cong \overline {XY}$	1. Given
2. $\angle {WZY} \cong \angle {XYZ}$	2. Base angles in an isosceles trapezoid are congruent
3. $\overline {ZY} \cong \overline {ZY}$	3. Reflexive Property.
4. $\triangle {WZY} \cong \triangle {XYZ}$	4. $SAS \cong SAS$
5. $\overline {WY} \cong \overline {XZ}$	5. Corresponding parts of congruent triangles are congruent

So, the two diagonals in the isosceles trapezoid are congruent. This will be true in any isosceles trapezoids.

Identifying Isosceles Trapezoids with Diagonals

The converse statement of the theorem stating that diagonals in an isosceles triangle are congruent is also true. If a trapezoid has congruent diagonals, it is an isosceles trapezoid. You can either use measurements shown on a diagram or use the distance formula to find the lengths. If you can prove that the diagonals are congruent, then you can identify the trapezoid as isosceles.

Theorem: If a trapezoid has congruent diagonals, then it is an isosceles trapezoid

Example 4

Is the trapezoid on the following grid isosceles?

It is true that you could find the lengths of the two sides to identify whether or not this trapezoid is isosceles. However, for the sake of this lesson, compare the lengths of the diagonals.

Begin by finding the length of $\overline{GJ}$ . The coordinates of $G$ are $(2,5)$ and the coordinates of $J$ are $(7,-1)$ .

$GJ &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\ &= \sqrt{(7-2)^2 + (-1-5)^2} \\ &= \sqrt{(5)^2 + (-6)^2} \\ &=\sqrt{25 + 36} \\ &= \sqrt{61}$

Now find the length of $\overline{HK}$ . The coordinates of $H$ are $(5,5)$ and the coordinates of $K$ are $(0,-1)$ .

$HK &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\ &= \sqrt{(0-5)^2 + ((-1)-5)^2} \\ &= \sqrt{(-5)^2 + (-6)^2} \\ &=\sqrt{25 + 36} \\ &= \sqrt{61}$

Thus, we have shown that the diagonals are congruent. $GJ=HK=\sqrt {61}$ . Therefore, trapezoid $GHJK$ is isosceles.

Trapezoid Medians

Trapezoids can also have segments drawn in called medians. The median of a trapezoid is a segment that connects the midpoints of the non-parallel sides in a trapezoid. The median is located half way between the bases of a trapezoid.

Example 5

In trapezoid $DEFG$ below, segment $XY$ is a median. What is the length of $\overline {EX}?$

The median of a trapezoid is a segment that is equidistant between both bases. So, the length of $\overline {EX}$ will be equal to half the length of $\overline {EF}$ . Since you know that $EF = 8\;\mathrm{inches}$ , you can divide that value by $2$ . Therefore, $XE$ is $4\;\mathrm{inches}$ .

Theorem: The length of the median of a trapezoid is equal to half of the sum of the lengths of the bases

This theorem can be illustrated in the example above,

$XY & = \frac{FG + ED} {2}\\ XY & = \frac{4 + 10} {2}\\ XY & = 7$

Therefore, the measure of segment $XY$ is $7\;\mathrm{inches}$ . We leave the proof of this theorem as an exercise, but it is similar to the proof that the length of the triangle midsegment is half the length of the base of the triangle.

Lesson Summary

In this lesson, we explored trapezoids. Specifically, we have learned to:

Understand and prove that the base angles of isosceles trapezoids are congruent.
Understand that if base angles in a trapezoid are congruent, it is an isosceles trapezoid.
Understand that the diagonals in an isosceles trapezoid are congruent.
Understand that if the diagonals in a trapezoid are congruent, the trapezoid is isosceles.
Identify the properties of the median of a trapezoid.

It is helpful to be able to identify specific properties in trapezoids. You will be able to use this information in many different ways.

Review Questions

Use the following figure for exercises 1-2.

$m \angle ADC =$
$m \angle BCD =$

Use the following figure for exercises 3-5.

$m \angle APR = 73^\circ$

$TP = 11.5 \;\mathrm{cm}$

$m \angle RAP =$
$AR =$ ________
$m \angle ATR =$

Use the following diagram for exercises 6-7.

$m \angle MAE =$
$EA =$
Can the parallel sides of a trapezoid be congruent? Why or why not? Use a sketch to illustrate your answer.
Can the diagonals of a trapezoid bisect each other? Why or why not? Use a sketch to illustrate your answer.
Prove that the length of the median of a trapezoid is equal to half of the sum of the lengths of the bases.

Review Answers

$40^\circ$
$140^\circ$
$17^\circ$
$11.5 \;\mathrm{cm}$
$107^\circ$
$84^\circ$
$18 \;\mathrm{cm}$
No, if the parallel (and by definition opposite) sides of a quadrilateral are congruent then the quadrilateral MUST be a parallelogram. When you sketch it, the two other sides must also be parallel and congruent to each other (proven in a previous section).
No, if the diagonals of a trapezoid bisect each other, then you have a parallelogram. We also proved this in a previous section.
We will use a paragraph proof.
Start with trapezoid $ABCD$ and midsegment $\overline{FE}$ .

Now, using the parallel postulate, construct a line through point $A$ that is parallel to $\overline{CD}$ . Label the new intersections as follows:

Now quadrilateral $AGCD$ is a parallelogram by construction. Thus, the theorem about opposite sides of a parallelogram tells us $AD = GC = HE.$ The triangle midsegment theorem tells us that

$FH = \frac{1} {2} BG$ or $BG = 2FH$

So,

$\frac{BC + AD} {2} & = \frac{BG + GC + AD} {2}\ && \text{by the segment addition postulate} \\ & = \frac{2FH + 2HE} {2} && \text{by substitution} \\ & = FH + HE && \text{by factoring out and canceling the}\ 2 \\ & = FE && \text{by the segment addition postulate. Which is exactly what we wanted to show!}$

Last modified: Wednesday, May 12, 2010, 4:37 PM