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.
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.
An isosceles trapezoid  its legs are congruent. 
These definitions result in the following properties of trapezoids:
This properties is true for all trapezoids:
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° 
The following properties are true for isosceles trapezoids.
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 

If a trapezoid is isosceles, then its diagonals are congruent.
$AC=BD$AC=BD 
We can use these properties to find unknown angles and sides of a trapezoid.
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.
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:
$x+113$x+113 
$=$=  $180$180 
$x$x  $=$=  $180113$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°
Consider the following trapezoid $JKLM$JKLM, where $KM=20$KM=20.
Find the value of $JL$JL.
Consider the following trapezoid, with $\overline{QR}\parallel\overline{TS}$QR∥TS.
Find the value of $x$x.
Find $m\angle Q$m∠Q.
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.
Trapezoid 
If the legs of the trapezoid are congruent, then it is an isosceles trapezoid.
Isosceles trapezoid 
If a trapezoid is isosceles, then it can be shown that:
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.
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 sideangleside 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 twocolumn proof where each line contains a geometric statement in the left column and a corresponding reason in the right column.
Given $ABCD$ABCD is an isosceles trapezoid, prove that its diagonals are congruent. 

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  Sideangleside 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.
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. 

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.
If a trapezoid has one pair of congruent base angles, then it is isosceles.
If a quadrilateral is a trapezoid, then base angles are supplementary.
If a quadrilateral has one pair of supplementary base angles, then it is a trapezoid.
If a trapezoid is isosceles, then base angles are congruent.
If a trapezoid has one pair of congruent base angles, then it is isosceles.
If a quadrilateral is a trapezoid, then base angles are supplementary.
If a quadrilateral has one pair of supplementary base angles, then it is a trapezoid.
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  Sideangleside congruence theorem 
$\triangle AED\cong\triangle BFC$△AED≅△BFC  Hypotenuseleg congruence theorem 
$\angle ADE\cong\angle BCF$∠ADE≅∠BCF  Corresponding parts of congruent triangles are congruent (CPCTC). 
$\triangle AED\cong\triangle BFC$△AED≅△BFC  Hypotenuseleg congruence theorem 
$\angle ADE\cong\angle BCF$∠ADE≅∠BCF  If a trapezoid is isosceles, then base angles are congruent. 
$\overline{AC}\cong\overline{BD}$AC≅BD  If a trapezoid is isosceles, then its diagonals are congruent. 
$\angle ABC\cong\angle CDA$∠ABC≅∠CDA  Sidesideside congruence theorem 
$\angle ADE\cong\angle BCF$∠ADE≅∠BCF  Corresponding parts of congruent triangles are congruent (CPCTC). 
$\triangle AED\cong\triangle BFC$△AED≅△BFC  Sideangleside congruence theorem 
$\triangle AED\cong\triangle BFC$△AED≅△BFC  Hypotenuseleg congruence theorem 
$\angle ADE\cong\angle BCF$∠ADE≅∠BCF  Corresponding parts of congruent triangles are congruent (CPCTC). 
$\triangle AED\cong\triangle BFC$△AED≅△BFC  Hypotenuseleg congruence theorem 
$\angle ADE\cong\angle BCF$∠ADE≅∠BCF  If a trapezoid is isosceles, then base angles are congruent. 
$\overline{AC}\cong\overline{BD}$AC≅BD  If a trapezoid is isosceles, then its diagonals are congruent. 
$\angle ABC\cong\angle CDA$∠ABC≅∠CDA  Sidesideside congruence theorem 
Identify and/or use properties of quadrilaterals