What if you were told that the polygon ABCD is an isosceles trapezoid and that one of its base angles measures

### 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

- Draw a trapezoid on your graph paper with vertices
A(−1,5), B(2,5), C(6,1) andD(−3,1) . Notice this is NOT an isosceles trapezoid. - Find the midpoint of the non-parallel sides either by using slopes or the midpoint formula. Label them
E andF . Connect the midpoints to create the midsegment. - Find the lengths of
AB, EF , andCD . 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

#### Measuring Angles

Look at trapezoid

Notice that

#### Writing a Two-Column Proof

Write a two-column proof.

Given: Trapezoid

Prove:

Statement |
Reason |
---|---|

1. Trapezoid |
Given |

2. |
Opposite Sides Theorem |

3. |
Corresponding Angles Postulate |

4. |
Transitive PoC |

5. |
Base Angles Converse |

6. |
Transitive PoC |

#### Solving for Unknown Values

Find

1. a)

b) 24 is the average of

c) 20 is the average of

#### Earlier Problem Revisited

Given an isosceles trapezoid with

In an isosceles trapezoid, base angles are congruent.

### Examples

Find:

#### Example 1

m∠TPA

#### Example 2

m∠PTR

#### Example 3

m∠PZA

By the Triangle Sum Theorem,

#### Example 4

m∠ZRA

Since \begin{align*}m\angle PZA = 110^\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.

- \begin{align*}A(-3, 2), B(1, 3), C(3, -1), D(-4, -2)\end{align*}
A(−3,2),B(1,3),C(3,−1),D(−4,−2) - \begin{align*}A(-3, 3), B(2, -2), C(-6, -6), D(-7, 1)\end{align*}
A(−3,3),B(2,−2),C(−6,−6),D(−7,1) - \begin{align*}A(1, 3), B(4, 0), C(2, -4), D(-3, 1)\end{align*}
A(1,3),B(4,0),C(2,−4),D(−3,1) - \begin{align*}A(1, 3), B(3, 1), C(2, -4), D(-3, 1)\end{align*}
A(1,3),B(3,1),C(2,−4),D(−3,1)

- Write a two-column proof of the Isosceles Trapezoid Diagonals Theorem using congruent triangles.

Given: \begin{align*}TRAP\end{align*}

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

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

### Review (Answers)

To view the Review answers, open this PDF file and look for section 6.6.