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5.13: Trapezoids

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  • 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 \(m\angle TPA\).


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

    Example \(\PageIndex{2}\)

    Find \(m\angle ZRA\).


    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 \(\PageIndex{7}\)


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

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

    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}

    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 \(\PageIndex{8}\)


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

    Example \(\PageIndex{5}\)

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

    1. f-d_75a8a92e617fe446439515147e845df56ed7391a470108614bd12810+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{10}\)
    1. f-d_7671594a2c974ac4555a516ea82ddf87c05a2a433848f47fce3853b0+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{10}\)
    2. f-d_caa24327458dc5f2f7c1aee9220318bd2822fff11c72f9f042517de3+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{11}\)


    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.

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

    1. 20 is the average of \(5x−15\) and \(2x−8\).

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


    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.

    1. f-d_8174a765e88593cc3813a93996d59d00778ea43c891df8dd62763f3c+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{12}\)
    2. f-d_6c1be5ea5cb93433e44642cbac4d3fc2078d65630b2bc67fdcfb642b+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{13}\)
    3. f-d_3f8a6e3e4e1dac669f1d9169b634ebd2a8eb4f0fd5b60c787c076e66+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{14}\)
    4. f-d_eae981cc91905c3158a1d484657596c6c91e4e9e9699af766999b969+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{15}\)
    5. f-d_54789fe8635cc70ebf85cebf2fd18981df3c414380d8a009a844a714+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{16}\)
    6. f-d_0f0366c36473d55c378e18f0788a63ad070a7205a5b3775c7b2c34f0+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{17}\)

    Find the value of the missing variable(s).

    1. f-d_74ac3cd5215d4ea2852c000bb03739aa1a53629c2628d9998d05c7c5+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{18}\)

    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)

    Review (Answers)

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


    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