# 2.12: Trapezoid Properties

**At Grade**Created by: CK-12

## Learning Objectives

- Identify and classify a
*trapezoid*. - Identify and classify an isosceles trapezoid.
- State that the base angles of isosceles trapezoids are congruent.
- State that if base angles in a trapezoid are congruent, it is an isosceles trapezoid.
- State that the diagonals in an isosceles trapezoid are congruent.
- State that if the diagonals in a trapezoid are congruent, the trapezoid is isosceles.
- Identify the median of a trapezoid and use its properties.

## Trapezoids

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.

**Trapezoids** have exactly one set of ______________________ sides.

**Trapezoids** are quadrilaterals but ____________ parallelograms.

There are special relationships in trapezoids, particularly in isosceles trapezoids.

Below is an example of the difference between isosceles and non-isosceles trapezoids:

*The word “isosceles” means “equal legs.”*

*Isosceles trapezoids have non-parallel sides that are of the same lengths.*

*These equal sides are sometimes called the “legs.”*

**Isosceles trapezoids** have *non-parallel sides* (called _________________ ) that are the same __________________________.

## Base Angles in Isosceles Trapezoids

The two angles along the same *base* in an isosceles triangle will be *congruent*. Thus, this creates two pairs of congruent angles—one pair along each base.

**Theorem for Isosceles Trapezoid**

The *base angles* of an **isosceles trapezoid** are *congruent*.

Each set of *base angles* in an **isosceles trapezoid** are _____________________________.

There are _____________ pairs of *base angles* in an **isosceles trapezoid**.

**Example 1**

*Examine trapezoid \begin{align*}ABCD\end{align*} below. What is the measure of angle \begin{align*}ADC\end{align*}?*

This problem requires two steps to solve.

Step 1: You already know that *base angles* in an **isosceles trapezoid** will be *congruent*, but you need to find the relationship between *adjacent* angles as well.

Imagine extending the parallel segments \begin{align*}\overline{BC}\end{align*} and \begin{align*}\overline{AD}\end{align*} on the trapezoid and the transversal \begin{align*}\overline{AB}\end{align*}. You will notice that the angle labeled \begin{align*}115^\circ\end{align*} is a *consecutive interior* angle with \begin{align*}\angle BAD\end{align*}.

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

\begin{align*}m \angle BAD + 115^\circ &= 180^\circ\\ m \angle BAD &= 65^\circ\end{align*}

So, \begin{align*}\angle BAD\end{align*} measures \begin{align*}65^\circ\end{align*}.

Step 2: Since \begin{align*}\angle BCD\end{align*} is *adjacent* to the same *base* as \begin{align*}\angle ADC\end{align*} in an **isosceles trapezoid**, the two angles must be *congruent*. Therefore, \begin{align*}m \angle ADC = 65^\circ\end{align*}.

## Identify Isosceles Trapezoids with Base Angles

You previously 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.

**Theorem for Trapezoid**

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

If base angles are *congruent*, then the trapezoid is __________________________.

You can use this fact to identify lengths in different trapezoids. An **isosceles trapezoid** has one pair of *congruent* sides:

**Example 2**

*What is the length of \begin{align*}\overline{MN}\end{align*} in the trapezoid below?*

Notice that in trapezoid \begin{align*}LMNO\end{align*}, 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 \begin{align*}\overline{MN}\end{align*}, it will be congruent to \begin{align*}\overline{LO}\end{align*}. So, \begin{align*}MN = 3\end{align*} feet.

**Reading Check:**

*Label as much information on the following isosceles trapezoid as you can.*

## 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 for Trapezoids Diagonals**

The *diagonals* of an **isosceles trapezoid** are *congruent*.

The *diagonals* in an **isosceles trapezoid** are ____________________________________.

**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. If you can prove that the diagonals are congruent, then you can identify the trapezoid as isosceles.

**Theorem for Trapezoid Diagonals**

If a trapezoid has *congruent diagonals*, then it is an **isosceles trapezoid**.

If a trapezoid has *diagonals* that are *congruent*, then it is __________________________.

**Example 3**

In the figure below, \begin{align*}DB = AC\end{align*}. What is the length of \begin{align*}\overline{AB}\end{align*}?

Because \begin{align*}DB\end{align*} and \begin{align*}AC\end{align*} are *diagonals* of trapezoid \begin{align*}ABCD\end{align*}, and \begin{align*}DB\end{align*} and \begin{align*}AC\end{align*} are *congruent*, we know that this trapezoid is **isosceles**.

- Isosceles trapezoids have two congruent sides.
- Since \begin{align*}CD = 4 \ cm\end{align*}, \begin{align*}AB\end{align*} must also be equal to 4 cm.

## 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.

A **median** connects the _______________________ of the *non-parallel* sides in a trapezoid.

A trapezoid’s **median** is half way between its ________________________.

**Example 4**

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

The **median** of a trapezoid is a segment that is *equidistant* between both *bases*. In other words, it divides the sides into two *congruent* parts.

- So, the length of \begin{align*}\overline{EX}\end{align*} will be equal to
*half*the length of \begin{align*}\overline{EF}\end{align*}. - Since you know that \begin{align*}EF = 8\end{align*} inches, you can divide that value by 2. Therefore, \begin{align*}EX\end{align*} is 4 inches.

**Theorem for Trapezoid Medians**

The length of the median of a trapezoid is equal to *half* of the *sum of the lengths of the bases*.

In other words, to find the length of the median, *average* the two bases.

Remember, the average is the *sum* of both numbers (bases) *divided* by 2.

This theorem can be illustrated in the example above,

\begin{align*}XY & = \frac{FG + ED}{2}\!\\ XY & = \frac{4 + 10}{2}\!\\ XY & = 7\end{align*}

Therefore, the measure of segment \begin{align*}XY\end{align*} is 7 inches.

- The length of a trapezoid’s ________________________ is the
*average*of its*bases*.

**Reading Check:**

*Find the following measures in trapezoid \begin{align*}ABCD\end{align*} below:*

\begin{align*}MN = \underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

\begin{align*}MA = \underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

\begin{align*}BA = \underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

\begin{align*}CD = \underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

\begin{align*}ND = \underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

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