Prove that the diagonals of an isosceles trapezoid are congruent
In order to prove that the diagonals of an isosceles trapezoid are congruent, consider the isosceles trapezoid shown below. In this lesson, we will show you two different ways you can do the same proof using the same trapezoid.
The first way is to show that triangle ABC is congruent to triangle DCB
Given: Isosceles trapezoid ABCD with segment AB ≅
segment DC
Prove: segment AC ≅ segment BD
Since the trapezoid is isosceles, the base angles are congruent.
Therefore, ∠CBA ≅ ∠BCD
Furthermore, segment BC ≅ to segment BC by the reflexive property of congruence.
By SAS Postulate, triangle ABC ≅ triangle DCB.
Therefore, segment AC ≅ segment BD
Things that you need to keep in mind when you prove that the diagonals of an isosceles trapezoid are congruent.
Here are some things that you must know about the proof above.
 The statement if a trapezoid is isosceles, then the base angles are congruent requires also a proof. However, we will not prove here.
 The sides that are equal in an isosceles trapezoid are always the sides that are not parallel. In the trapezoid above, we show these sides with the red marks.
 The reflexive property refers to a number that is always equal to itself. For example, 9 = 9 or y = y are examples of the reflexive property.
 SAS stands for "side, angle, side". You should perhaps review the lesson about congruent triangles.
 In order to prove that the diagonals of an isosceles trapezoid are congruent, you could have also used triangle ABD and triangle DCA.
Another great way to prove that the diagonals of an isosceles trapezoid are congruent. This time we show that triangle BAD is congruent to triangle CDA
Given: Isosceles trapezoid ABCD with segment AB ≅
segment DC
Prove: segment AC ≅ segment BD
Since the trapezoid is isosceles, the base angles are congruent
Therefore, ∠BAD ≅ ∠CDA
Notice that this time, we are not using the same base angles as before. The base angles we are using now is related to the base on top or segment AD.
Furthermore, segment AD ≅ to segment AD by the reflexive property of congruence.
By SAS Postulate, triangle BAD ≅ triangle CDA
Therefore, segment AC ≅ segment BD

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