5.13: Trapezoids

Determine unknown angle measurements of quadrilaterals with exactly one pair of parallel sides.

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

Figure \(\PageIndex{1}\)

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

Figure \(\PageIndex{2}\)

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

Figure \(\PageIndex{3}\)

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:

Figure \(\PageIndex{4}\)

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.

Figure \(\PageIndex{5}\)

If \(\overline{EF}\) is the midsegment, then \(EF=\dfrac{AB+CD}{2}\).

What if you were told that the polygon \(ABCD\) is an isosceles trapezoid and that one of its base angles measures \(38^{\circ}\)? What can you conclude about its other base angle?

For Examples 1 and 2, use the following information:

\(\(TRAP\)\) is an isosceles trapezoid.

Figure \(\PageIndex{6}\)

Example \(\PageIndex{1}\)

Find \(m\angle TPA\).

Solution

\(\angle TPZ\cong \angle RAZ\) so \(m\angle TPA=20^{\circ} +35^{\circ} =55^{\circ}\).

Example \(\PageIndex{2}\)

Find \(m\angle ZRA\).

Solution

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

Example \(\PageIndex{3}\)

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

Figure \(\PageIndex{7}\)

Solution

\(TRAP\) is an isosceles trapezoid. \(m\angle R=115^{\circ}\) also.

To find \(m\angle , set up an equation.

\(\begin{aligned}
115^{\circ}+115^{\circ}+m \angle A+m \angle P &=360^{\circ} \\
230^{\circ}+2 m \angle A &=360^{\circ} \quad \rightarrow m \angle A=m \angle P \\
2 m \angle A &=130^{\circ} \\
m \angle A &=65^{\circ}
\end{aligned}\)

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 \(\PageIndex{4}\)

Is \(ZOID\) an isosceles trapezoid? How do you know?

Figure \(\PageIndex{8}\)

Solution

\(40^{\circ} \neq 35^{\circ}\), \(ZOID\) is not an isosceles trapezoid.

Example \(\PageIndex{5}\)

Find \(x\). All figures are trapezoids with the midsegment marked as indicated.

1. 

   Figure \(\PageIndex{10}\)

2. 

   Figure \(\PageIndex{10}\)

3. 

   Figure \(\PageIndex{11}\)

Solution

1. \(x\) is the average of 12 and 26. \(\dfrac{12+26}{2}=\dfrac{38}{2}=19\)
2. 24 is the average of \(x\) and 35.

   \(\begin{aligned} \dfrac{x+35}{2}&=24 \\ x+35&=48 \\ x&=13 \end{aligned}\)

3. 20 is the average of \(5x−15\) and \(2x−8\).

   \(\begin{aligned} \dfrac{5x−15+2x−8}{2}&=20 \\ 7x−23&=40 \\ 7x&=63 \\ x&=9 \end{aligned}\)

Review

1. Can the parallel sides of a trapezoid be congruent? Why or why not?

For questions 2-8, find the length of the midsegment or missing side.

2. 

   Figure \(\PageIndex{12}\)

3. 

   Figure \(\PageIndex{13}\)

4. 

   Figure \(\PageIndex{14}\)

5. 

   Figure \(\PageIndex{15}\)

6. 

   Figure \(\PageIndex{16}\)

7. 

   Figure \(\PageIndex{17}\)

Find the value of the missing variable(s).

8. 

   Figure \(\PageIndex{18}\)

Find the lengths of the diagonals of the trapezoids below to determine if it is isosceles.

9. A(−3,2), B(1,3), C(3,−1), D(−4,−2)
10. A(−3,3), B(2,−2), C(−6,−6), D(−7,1)

Review (Answers)

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

Vocabulary

Term	Definition
isosceles trapezoid	An isosceles trapezoid is a trapezoid where the non-parallel sides are congruent.
midsegment (of a trapezoid)	A line segment that connects the midpoints of the non-parallel sides.
trapezoid	A quadrilateral with exactly one pair of parallel sides.
Diagonal	A diagonal is a line segment in a polygon that connects nonconsecutive vertices
midsegment	A midsegment connects the midpoints of two sides of a triangle or the non-parallel sides of a trapezoid.

Additional Resources

Interactive Element

Video: Trapezoids Examples - Basic

Activities: Trapezoids Discussion Questions

Study Aids: Trapezoids and Kites Study Guide

Practice: Trapezoids

Real World: Trapezoids in Timbuktu