# Prove that the diagonals of a rectangle are congruent

In order to prove that the diagonals of a rectangle are congruent, consider the rectangle shown below. In this lesson, we will show you two different ways you can do the same proof using the same rectangle. ## The first way to prove that the diagonals of a rectangle are congruent is to show that triangle ABC is congruent to triangle DCB

Here is what is given: Rectangle ABCD

Here is what you need to prove: segment AC ≅ segment BD

Since ABCD is a rectangle, it is also a parallelogram.

Since ABCD is a parallelogram, segment AB ≅ segment DC because opposite sides of a parallelogram are congruent.

BC ≅ BC by the Reflexive Property of Congruence.

Furthermore, ∠ABC and ∠DCB are right angles by the definition of rectangle.

∠ABC ≅ ∠DCB since all right angles are congruent.

Summary

segment AB ≅ segment DC

∠ABC ≅ ∠DCB

BC ≅ BC

Therefore, by SAS, triangle ABC ≅ triangle DCB.

Since triangle ABC ≅ triangle DCB, segment AC ≅ segment BD

## Things that you need to keep in mind when you prove that the diagonals of a rectangle are congruent.

Here are some important things that you should be aware of about the proof above.

• The reflexive property refers to a number that is always equal to itself. For example, x = x or -6 = -6 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 a rectangle are congruent, you could have also used triangle ABD and triangle DCA.

## The second way to prove that the diagonals of a rectangle are congruent is to show that triangle ABD is congruent to triangle DCA

Here is what is given: Rectangle ABCD

Here is what you need to prove: segment AC ≅ segment BD

Since ABCD is a rectangle, it is also a parallelogram.

Since ABCD is a parallelogram, segment AB ≅ segment DC because opposite sides of a parallelogram are congruent.

Furthermore, ∠BAD and ∠CDA are right angles by the definition of rectangle.

∠BAD ≅ ∠CDA since all right angles are congruent.

Summary

segment AB ≅ segment DC

Therefore, by SAS, triangle ABD ≅ triangle DCA.

Since triangle ABD ≅ triangle DCB, segment AC ≅ segment BD

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