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# Trapezoids

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Trapezoids

What if you were told that the polygon $ABCD$ is an isoceles trapezoid and that one of its base angles measures $38^\circ$ ? What can you conclude about its other base angle? After completing this Concept, you'll be able to find the value of a trapezoid's unknown angles and sides.

### Guidance

A trapezoid is a quadrilateral with exactly one pair of parallel sides.

An isosceles trapezoid is a trapezoid where the non-parallel sides are congruent.

The base angles of an isosceles trapezoid are congruent. If $ABCD$ is an isosceles trapezoid, then $\angle A \cong \angle B$ and $\angle C \cong \angle D$ .

The converse is also true. If a trapezoid has congruent base angles, then it is an isosceles trapezoid. The diagonals of an isosceles trapezoid are also 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.

Midsegment Theorem: The length of the midsegment of a trapezoid is the average of the lengths of the bases.

If $\overline{EF}$ is the midsegment, then $EF = \frac{AB + CD}{2}$ .

#### Example A

Look at trapezoid $TRAP$ below. What is $m \angle A$ ?

$TRAP$ is an isosceles trapezoid. $m \angle R = 115^\circ$ also.

To find $m \angle A$ , set up an equation.

$115^\circ + 115^\circ + m \angle A + m \angle P & = 360^\circ\\230^\circ + 2m \angle A & = 360^\circ \qquad \rightarrow m \angle A = m \angle P\\2m \angle A & = 130^\circ\\m \angle A & = 65^\circ$

Notice that $m \angle R + m \angle A = 115^\circ + 65^\circ = 180^\circ$ . These angles will always be supplementary because of the Consecutive Interior Angles Theorem.

#### Example B

Is $ZOID$ an isosceles trapezoid? How do you know?

$40^\circ \neq 35^\circ$ , $ZOID$ is not an isosceles trapezoid.

#### Example C

Find $x$ . All figures are trapezoids with the midsegment marked as indicated.

a)

b)

c)

a) $x$ is the average of 12 and 26. $\frac{12 + 26}{2} = \frac{38}{2} = 19$

b) 24 is the average of $x$ and 35.

$\frac{x + 35}{2} & = 24\\x + 35 & = 48\\x & = 13$

c) 20 is the average of $5x - 15$ and $2x - 8$ .

$\frac{5x - 15 + 2x - 8}{2} & = 20\\7x - 23 & = 40\\7x & = 63\\x & = 9$

### Guided Practice

$TRAP$ an isosceles trapezoid.

Find:

1. $m \angle TPA$
2. $m \angle PTR$
3. $m \angle PZA$
4. $m \angle ZRA$

1. $\angle TPZ \cong \angle RAZ$ so $m\angle TPA=20^\circ + 35^\circ = 55^\circ$ .

2. $\angle TPA$ is supplementary with $\angle PTR$ , so $m\angle PTR=125^\circ$ .

3. By the Triangle Sum Theorem, $35^\circ + 35^\circ + m\angle PZA=180^\circ$ , so $m\angle PZA=110^\circ$ .

4. Since $m\angle PZA = 110^\circ$ , $m\angle RZA=70^\circ$ because they form a linear pair. By the Triangle Sum Theorem, $m\angle ZRA=90^\circ$ .

### Practice

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. $A(-3, 2), B(1, 3), C(3, -1), D(-4, -2)$
2. $A(-3, 3), B(2, -2), C(-6, -6), D(-7, 1)$

### Vocabulary Language: English Spanish

isosceles trapezoid

isosceles trapezoid

An isosceles trapezoid is a trapezoid where the non-parallel sides are congruent.
midsegment (of a trapezoid)

midsegment (of a trapezoid)

A line segment that connects the midpoints of the non-parallel sides.
trapezoid

trapezoid

A quadrilateral with exactly one pair of parallel sides.
Diagonal

Diagonal

A diagonal is a line segment in a polygon that connects nonconsecutive vertices
midsegment

midsegment

A midsegment connects the midpoints of two sides of a triangle or the non-parallel sides of a trapezoid.