11.04 Properties of trapezoids
Solve for missing sides or angles in trapezoids
A trapezoid is a quadrilateral where exactly one pair of its sides are parallel. To examine the trapezoid we classify different parts of it. The two parallel sides are called bases. The other two sides, which are not parallel, are called legs. The base angle pairs are the two angles between a leg and the same base.
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| A trapezoid - it has only one pair of parallel sides. | The legs and bases of a trapezoid. |
A trapezoid is isosceles if its two legs are congruent.
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| An isosceles trapezoid - its legs are congruent. |
These definitions result in the following properties of trapezoids:
This properties is true for all trapezoids:
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If a quadrilateral is a trapezoid, then consecutive interior angles of the parallel bases will be supplementary ($x+y=180$x+y=180).
$m\angle A+m\angle D=180^\circ$m∠A+m∠D=180° $m\angle B+m\angle C=180^\circ$m∠B+m∠C=180° |
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The following properties are true for isosceles trapezoids.
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If a trapezoid is isosceles, then each pair of base angles are congruent.
$m\angle A=m\angle B$m∠A=m∠B $m\angle C=m\angle D$m∠C=m∠D |
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If a trapezoid is isosceles, then its diagonals are congruent.
$AC=BD$AC=BD |
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We can use these properties to find unknown angles and sides of a trapezoid.
Worked examples
QUestion 1
Consider the trapezoid below. The length of $\overline{AC}$AC is $9$9 units.
Find the length of $\overline{BD}$BD.
Think: The two legs of the trapezoid are equal so it is an isosceles trapezoid. The diagonals of an isosceles trapezoid are equal.
Do: The length of $\overline{AC}$AC will be equal to $\overline{BD}$BD. So $BD=9$BD=9.
Question 2
Find the values of $x$x and $y$y in the trapezoid below.
Think: This is an isosceles trapezoid and the angles marked $\angle C$∠C and $\angle D$∠D are a base angle pair so they will be equal.
The angles $\angle A$∠A and $\angle D$∠D are consecutive interior angles of the parallel bases, so they are supplementary (their measures will sum to $180^\circ$180°).
Do: The two supplementary angles form the equation:
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$x+113$x+113 |
$=$= | $180$180 |
| $x$x | $=$= | $180-113$180−113 |
| $x$x | $=$= | $67$67 |
The angle $\angle C$∠C will be congruent to $\angle D$∠D so $x=y=67$x=y=67.
Reflect: The remaining unknown angle is $\angle B$∠B. This angle is congruent to $\angle A$∠A and supplementary with $\angle C$∠C. Therefore $m\angle B=113^\circ$m∠B=113°
Practice questions
Question 3
Consider the following trapezoid $JKLM$JKLM, where $KM=20$KM=20.
Find the value of $JL$JL.
question 4
Consider the following trapezoid, with $\overline{QR}\parallel\overline{TS}$QR∥TS.
Trapezoid $QRST$QRST is drawn with given two interior angles. The given interior angles are two consecutive angles in the trapezoid. $\angle RQT$∠RQT measures $\left(6x-22\right)$(6x−22) and $\angle QTS$∠QTS measures $\left(8x+34\right)$(8x+34).
Find the value of $x$x.
Find $m\angle Q$m∠Q.
Proving properties of trapezoids
A trapezoid is defined as a quadrilateral with exactly one pair of parallel sides. These sides are known as the bases of the trapezoid. The remaining two nonparallel sides are the legs. A base angle is formed by each base with each leg.
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| Trapezoid |
If the legs of the trapezoid are congruent, then it is an isosceles trapezoid.
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| Isosceles trapezoid |
If a trapezoid is isosceles, then it can be shown that:
- each pair of base angles are congruent
- its diagonals are congruent
Exploration
Consider the following theorem.
If a trapezoid is isosceles, then its diagonals are congruent.
To begin proving the above theorem, we can first label the trapezoid by its set of vertices, $ABCD$ABCD, and draw two additional segments representing the diagonals, $\overline{AC}$AC and $\overline{BD}$BD.
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| Trapezoid $ABCD$ABCD with diagonals $\overline{AC}$AC and $\overline{BD}$BD. |
From the definition of an isosceles trapezoid, we know that the two legs are congruent. We can write this as a geometric statement, $\overline{AD}\cong\overline{BC}$AD≅BC.
Next, we can use the fact that each pair of base angles in an isosceles trapezoid are congruent to state that $\angle ADC\cong\angle BCD$∠ADC≅∠BCD.
Finally, we notice that the segment $\overline{DC}$DC is common to $\triangle ADC$△ADC and $\triangle BCD$△BCD. Clearly a segment is congruent to itself, so we can use the reflexive property of congruence we can state that $\overline{DC}\cong\overline{DC}$DC≅DC.
Hence, $\triangle ADC$△ADC and $\triangle BCD$△BCD have side-angle-side congruence. This now allows us to state that the diagonal $\overline{AC}$AC is congruent to the diagonal $\overline{BD}$BD, since they are corresponding parts of the two congruent triangles.
We can formalize the above steps into a two-column proof where each line contains a geometric statement in the left column and a corresponding reason in the right column.
Two-column proof
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Given $ABCD$ABCD is an isosceles trapezoid, prove that its diagonals are congruent.
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| Statements | Reasons |
| $ABCD$ABCD is an isosceles trapezoid | Given |
| $\overline{AD}\cong\overline{BC}$AD≅BC | Definition of an isosceles trapezoid |
| $\overline{DC}\cong\overline{DC}$DC≅DC | Reflexive property of congruence |
| $\angle ADC\cong\angle BCD$∠ADC≅∠BCD | If a trapezoid is isosceles, then each pair of base angles is congruent. |
| $\triangle ADC\cong\triangle BCD$△ADC≅△BCD | Side-angle-side congruence theorem |
| $\overline{AC}\cong\overline{BD}$AC≅BD | Corresponding parts of congruent triangles are congruent (CPCTC). |
The final line contains the statement that the two diagonals of the isosceles trapezoid are congruent, which is what we wanted to prove.
Practice questions
Question 5
Given the proof below, which of the following statements is shown to be true?
Given $\overline{AB}\parallel\overline{DC}$AB∥DC, $\overline{AD}\cong\overline{BC}$AD≅BC, and $\overline{AD}\parallel\overline{BE}$AD∥BE, prove that $\angle BCE\cong\angle ADE$∠BCE≅∠ADE and $\angle BAD\cong\angle ABC$∠BAD≅∠ABC. Note that $D$D, $E$E and $C$C are collinear.
A trapezoid $ABCD$ABCD is depicted. Point $E$E is on side $CD$CD of the trapezoid. A line is drawn from point $B$B to point $E$E, forming line segment $BE$BE. Side $AB$AB and $DC$DC are parallel. Side $AD$AD and $BC$BC are congruent. Side $AD$AD is parallel with side $BE$BE. Statements Reasons $\overline{AB}\parallel\overline{DC}$AB∥DC, $\overline{AD}\cong\overline{BC}$AD≅BC, and $\overline{AD}\parallel\overline{BE}$AD∥BE
Given $ABED$ABED is a parallelogram Definition of a parallelogram $\overline{AD}\cong\overline{BE}$AD≅BE If a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent. $\overline{BE}\cong\overline{BC}$BE≅BC Transitive property of congruence $\triangle EBC$△EBC is isosceles Definition of an isosceles triangle $\angle BCE\cong\angle BEC$∠BCE≅∠BEC If a triangle is isosceles, then its base angles are congruent. $\angle BEC\cong\angle ADE$∠BEC≅∠ADE If two lines are parallel, then corresponding angles are congruent (corresponding angles theorem) $\angle BCE\cong\angle ADE$∠BCE≅∠ADE Transitive property of congruence $\angle BCE$∠BCE and $\angle ABC$∠ABC are supplementary If two lines are parallel, then consecutive interior angles are supplementary (consecutive interior angles postulate) $\angle ADE$∠ADE and $\angle BAD$∠BAD are supplementary If two lines are parallel, then consecutive interior angles are supplementary (consecutive interior angles postulate) $\angle BAD\cong\angle ABC$∠BAD≅∠ABC Two angles supplementary to the same angle (or congruent angles) are congruent (congruent supplements theorem) If a trapezoid is isosceles, then base angles are congruent.
AIf a trapezoid has one pair of congruent base angles, then it is isosceles.
BIf a quadrilateral is a trapezoid, then base angles are supplementary.
CIf a quadrilateral has one pair of supplementary base angles, then it is a trapezoid.
D
Question 6
Given the proof below, select the correct statements and reasons.
Given $ABCD$ABCD is an isosceles trapezoid with altitudes $\overline{AE}$AE and $\overline{BF}$BF, show that each pair of base angles are congruent.
Statements Reasons $ABCD$ABCD is an isosceles trapezoid Given $\overline{AD}\cong\overline{BC}$AD≅BC Definition of an isosceles trapezoid $\overline{AE}\cong\overline{BF}$AE≅BF Altitudes of a trapezoid are congruent $\angle DEA$∠DEA and $\angle CFB$∠CFB are right angles Definition of an altitude $\angle DEA\cong\angle CFB$∠DEA≅∠CFB All right angles are congruent $\left[\text{_____}\right]$[_____] $\left[\text{_____}\right]$[_____] $\left[\text{_____}\right]$[_____] $\left[\text{_____}\right]$[_____] $\angle BCD$∠BCD and $\angle CBA$∠CBA are supplementary If two lines are parallel, then consecutive interior angles are supplementary (consecutive interior angles postulate) $\angle ADC$∠ADC and $\angle BAD$∠BAD are supplementary If two lines are parallel, then consecutive interior angles are supplementary (consecutive interior angles postulate) $\angle BAD\cong\angle ABC$∠BAD≅∠ABC Two angles supplementary to the same angle (or congruent angles) are congruent (congruent supplements theorem) $\angle ADE\cong\angle BCF$∠ADE≅∠BCF Corresponding parts of congruent triangles are congruent (CPCTC). $\triangle AED\cong\triangle BFC$△AED≅△BFC Side-angle-side congruence theorem A$\triangle AED\cong\triangle BFC$△AED≅△BFC Hypotenuse-leg congruence theorem $\angle ADE\cong\angle BCF$∠ADE≅∠BCF Corresponding parts of congruent triangles are congruent (CPCTC). B$\triangle AED\cong\triangle BFC$△AED≅△BFC Hypotenuse-leg congruence theorem $\angle ADE\cong\angle BCF$∠ADE≅∠BCF If a trapezoid is isosceles, then base angles are congruent. C$\overline{AC}\cong\overline{BD}$AC≅BD If a trapezoid is isosceles, then its diagonals are congruent. $\angle ABC\cong\angle CDA$∠ABC≅∠CDA Side-side-side congruence theorem D