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# 2.12: Trapezoid Properties

Difficulty Level: 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.

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.

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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