Isosceles Trapezoids
An isosceles trapezoid is a trapezoid whose legs are congruent.
Go to Geogebra: Isosceles Trapezoids to find out more.
Isosceles Trapezoids Theorems
If a trapezoid is isosceles, then each pair of base angles is congruent.
If ABDC is a trapezoid and AC ≅ BD, then ∠A ≅ ∠B and ∠C ≅ ∠D.
You can use this theorem to find the measure of missing angles of a trapezoid, knowing that the interior angles of a quadrilateral sum to 360∘.
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.
If ABDC is a trapezoid and ∠A ≅ ∠B and ∠C ≅ ∠D, then ABDC is an isosceles trapezoid and AC ≅ BD.
This is the converse of the previous theorem. It allows you to prove a trapezoid isosceles given the measures of its base angles. You can also use this theorem to find the length of a missing leg of a trapezoid, given the length of the other leg and congruent base angles.
A trapezoid is isosceles if and only if its diagonals are congruent.
If ABDC is a trapezoid and AD ≅ BC, then ABDC is an isosceles trapezoid.
This theorem can be used to prove trapezoids are isosceles or to find the length of missing diagonals.
Example #1
Find the angle measures of ABCD.
We can see from the information given in the diagram that the legs AD and BC are congruent, so trapezoid ABCD is an isosceles trapezoid. Knowing this, we are able to determine that since m∠A is 53∘, m∠B = 53∘ because they are both base angles.
∠A and ∠D are consecutive angles are supplementary, so they have a sum of 180∘ when added together. We can use this information to find the measure of the unknown angle.
m∠A + m∠D = 180 This is the definition of supplementary angles ∘
53 Substitute in the value for A ∘ + m∠D = 180∘
53∘ + m∠D - 53∘ = 180° - 53 Substract 53° from each side to isolate the unknown. ∘
m∠D = 127∘
∠D and ∠C are also base angles so they must also be congruent Remember the theorem we learned earlier? If a trapezoid is isosceles, than its base angle pairs are congruent. . Therefore, mC = 127∘.
Your Turn!
Find the angle measures of ABCD.
Based on the given information, this trapezoid is not an isosceles trapezoid, but we should still know how to solve for the missing angle measures. A and D are consecutive interior angles so they have a sum of 180∘ when added.
m∠A + m∠D = ____∘
Answer: 180
Substitute in our known values.
____∘ + m∠D = 180∘
Answer: 91
Isolate the unknown value.
m∠D = ____∘
Answer: 89
Likewise, ∠B and ∠C are consecutive interior angles so they have a sum of 180∘ when added.
m∠B + m∠C = ____∘
Answer: 180
Substitute in our known angles.
___ ∘ + m∠C = 180∘
Answer: 132
Isolate our unknown value.
m∠C = ___∘
Answer: 48
