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9. Polygons
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Geometry

9.04 Trapezoids

Lesson
Practice

Trapezoids

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

A trapezoid labeled with its parts. Ask your teacher for more information.

The parallel sides are called bases and the nonparallel sides are called legs.

Consecutive angles in a trapezoid whose common side is a base are base angles. A trapezoid has two pairs of base angles.

In this case, \angle 1 and \angle 2 are base angles to the top base and \angle 3 and \angle 4 are base angles to the bottom base.

An isosceles trapezoid is a quadrilateral with one set of opposite sides parallel and the other set of opposite sides congruent.

The following theorems are related to isosceles trapezoids:

Isosceles trapezoid diagonals theorem

If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.

An isosceles trapezoid with its 2 diagonals marked congruent.
Isosceles trapezoid base angles theorem

If a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent.

An isosceles trapezoid. Each pair of base angles is marked congruent.

A midsegment of a trapezoid is a line segment that bisects both legs.

Trapezoid midsegment theorem

The midsegment of a trapezoid is parallel to each base and the length of the midsegment is equal to the sum of the length of the bases divided by two.

\overline{PS} \parallel \overline{MN}, \overline{MN}\parallel \overline{QR}

\text{and}

\text{Length of midsegment}=\dfrac{b_1+b_2}{2}

\text{or}

\text{Length of midsegment}=\dfrac{1}{2}\left(b_1+b_2\right)

Trapezoid P Q R S is drawn with top base Q R and bottom base P S. Ask your teacher for more information.

Examples

Example 1

ABCD is a trapezoid.

Trapezoid A B C D with segment AB and segment D C marked as parallel. Ask your teacher for more information.

Solve for x.

Worked Solution
Create a strategy

Use the trapezoid midsegment theorem: \text{Length of midsegment}=\dfrac{b_1+b_2}{2}, where \\b_{1} = 3x+2,\,b_{2}=2x-2 and \text{midsegment length}=15.

Apply the idea
\displaystyle 15\displaystyle =\displaystyle \dfrac{\left(3x+2\right)+\left(2x-2\right)}{2}Substitute the values of base and midsegment lengths
\displaystyle 15\displaystyle =\displaystyle \dfrac{5x}{2}Combine like terms in the numerator
\displaystyle 30\displaystyle =\displaystyle 5xMultiply both sides of equation by 2
\displaystyle 6\displaystyle =\displaystyle xDivide both sides of equation by 5
\displaystyle x\displaystyle =\displaystyle 6Symmetric property of equality

Example 2

Given: Trapezoid PQRS

Trapezoid P Q R S is drawn in which side P S and Side Q R are marked congruent. Ask your teacher for more information.

Solve for x.

Worked Solution
Create a strategy

Combine the measurements of all the angles, and remember that all quadrilaterals' interior angles sum to 360 \degree.

Apply the idea
\displaystyle 2\left(m\angle PQR\right) + 2\left(m\angle PSR\right)\displaystyle =\displaystyle 360
\displaystyle 2\left(m\angle PQR + m\angle PSR\right)\displaystyle =\displaystyle 360Factor out the common factor of 2
\displaystyle 2\left(12x+9+63\right)\displaystyle =\displaystyle 360Substitute m\angle PQR=12x+9 and m\angle PSR = 63
\displaystyle \left(12x+9+63\right)\displaystyle =\displaystyle 180Divide both sides of equation by 2
\displaystyle 12x+72\displaystyle =\displaystyle 180Combine like terms
\displaystyle 12x\displaystyle =\displaystyle 108Subtract 72 from both sides of equation
\displaystyle x\displaystyle =\displaystyle 9Divide both sides of the equation by 12
Reflect and check

Bases of a trapezoid are parallel so \angle QPS and \angle PSR are supplementary using the consecutive interior angles theorem. We could have also used theorems of parallel lines to solve for x.

Example 3

ABFE is a trapezoid.

Trapezoid A E F D with midpoints C of side A E, and D of side B F. Ask your teacher for more information.
a

Explain whether or not m\angle A=73 \degree and m\angle F=117 \degree would make this diagram valid or not.

Worked Solution
Create a strategy

Notice that \overline{AE} \cong \overline{BF}, so this is an isosceles trapezoid. This means that the base angles should be congruent, so we need m\angle A = m\angle B=73 \degree and m\angle F=m\angle E=117 \degree for it to be valid.

Apply the idea

Since ABFE is a trapezoid \overline{AB} \parallel \overline{EF}, so using \overline{AE} as a transversal we get that \angle A and \angle E are supplementary. Similarly, using \overline{BF} as a transversal we get that \angle B and \angle F are supplementary.

If \angle A and \angle E are supplementary, then m \angle E=107 \degree.

If \angle B and \angle F are supplementary, then m \angle B=63 \degree.

Since both of these violate the requirement for base angles to be congruent, m\angle A=73 \degree and m\angle F=117 \degree makes this diagram invalid.

b

Solve for x.

Worked Solution
Create a strategy

Use the trapezoid midsegment theorem: \text{Length of midsegment}=\dfrac{b_1+b_2}{2}, where \\b_{1} = 140,\,b_{2}=10x-10 and \text{midsegment length}=8x+2.

Apply the idea
\displaystyle 8x+2\displaystyle =\displaystyle \dfrac{140+\left(10x-10\right)}{2}Substitution
\displaystyle 16x+4\displaystyle =\displaystyle 140+\left(10x-10\right)Multiply both sides by 2
\displaystyle 16x+4\displaystyle =\displaystyle 10x+130Combine like terms
\displaystyle 6x\displaystyle =\displaystyle 126Subtract 4 and 10x from both sides
\displaystyle x\displaystyle =\displaystyle 21Divide both sides by 6

Example 4

Given: Trapezoid PQRS

  • PR=4x-1
  • QS=2x+8
Trapezoid P Q R S with diagonals P R and Q S. Sides P Q and R S are congruent.

Solve for x.

Worked Solution
Create a strategy

The isosceles trapezoid diagonals theorem tells us that if a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.

Apply the idea

The markings tell us that this is an isosceles trapezoid, so PR=QS. We can use this to solve for x.

\displaystyle PR\displaystyle =\displaystyle QSIsosceles trapezoid diagonals theorem
\displaystyle 4x-1\displaystyle =\displaystyle 2x+8Substitute
\displaystyle 4x\displaystyle =\displaystyle 2x+9Add 1 to both sides
\displaystyle 2x\displaystyle =\displaystyle 9Subtract 2x from both sides
\displaystyle x\displaystyle =\displaystyle \dfrac{9}{2}Divide both sides by 2

Example 5

Prove the trapezoid midsegment theorem: the midsegment of a trapezoid is parallel to each base and the length of the midsegment is equal to the sum of the length of the bases divided by two.

Worked Solution
Create a strategy

Given: Trapezoid ABCD

  • E is the midpoint of \overline{AD}

  • F is the midpoint of \overline{BC}

We will draw an auxillary point G to help with this proof.

Trapezoid D A B C with points E on side D A and F on side B C. Ask your teacher for more information.
Apply the idea
To prove: \overline{EF} || \overline{DG}, EF=\dfrac{1}{2}(AB+DC)
StatementsReasons
1.Trapezoid ABCDGiven
2. E is the midpoint of \overline{AD} Given
3. F is the midpoint of \overline{BC} Given
4.\overline{AB} || \overline{DG} Definition of trapezoid
5.\angle BAF \cong \angle CGFAlternate interior angles theorem
6.\angle AED \cong \angle CEFVertical angles congruence theorem
7.\overline{BF} \cong \overline{CF}Definition of midpoint
8.\triangle ABF \cong \triangle GCFAngle-Side-Angle congruency theorem
9.\overline{AF} \cong \overline{FG}Corresponding parts of congruent triangles
10.\overline{EF} is a midsegmentDefinition of midsegment
11.\overline{EF} || \overline{DG}Triangle midsegment theorem
12.EF=\dfrac{1}{2}DGTriangle midsegment theorem
13.DG=DC+CG Segment addition postulate
14. CG=AB Corresponding parts of congruent triangles
15.EF=\dfrac{1}{2}(DC+CG) Substitution property
16.EF=\dfrac{1}{2}(AB+DC) Substitution property

Example 6

Use geometric constructions and properties of trapezoids to verify that quadrilateral PQRS is a trapezoid given that its non-parallel sides are \overline{PQ} and \overline{RS}.

Worked Solution
Create a strategy

If a quadrilateral has at least one pair of parallel sides, it is a trapezoid. So, we may use a parallel line construction to verify that one pair of opposite sides is indeed parallel.

Apply the idea

1. Choose a point T on \overline{QR} through which to construct a line parallel to \overline{PS}.

2. Set the compass width to the distance ST.

3. Construct an arc centered at P with the radius ST.

4. Set the compass width to the distance PT.

5. Construct an arc centered at S with the radius PT.

6. Construct a line through T and the intersection of the two arcs, labeling it U. This line is parallel to \overline{PS}.

Since \overleftrightarrow{TU} is parallel to \overline{PS} and coincides with \overline{QR}, we have shown that one pair of opposite sides of the quadrilateral are parallel and thus the quadrilateral is a trapezoid.

Idea summary

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

An isoscelese trapezoid has congruent legs, base angles, and diagonals.

\text{Length of midsegment}=\dfrac{b_{1}+b_{2}}{2} or \text{Length of midsegment}=\dfrac{1}{2}\left(b_{1} + b_{2}\right)

Outcomes

G.PC.1

The student will prove and justify theorems and properties of quadrilaterals, and verify and use properties of quadrilaterals to solve problems, including the relationships between the sides, angles, and diagonals.

G.PC.1a

Solve problems, using the properties specific to parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids.

G.PC.1b

Prove and justify that quadrilaterals have specific properties, using coordinate and algebraic methods, such as the slope formula, the distance formula, and the midpoint formula.

G.PC.1c

Prove and justify theorems and properties of quadrilaterals using deductive reasoning.

G.PC.1d

Use congruent segment, congruent angle, angle bisector, perpendicular line, and/or parallel line constructions to verify properties of quadrilaterals.

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