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.

Diagonals of a rectangle are congruent

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.

AD ≅ AD by the Reflexive Property of Congruence.

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

∠BAD ≅ ∠CDA

AD ≅ AD

Therefore, by SAS, triangle ABD ≅ triangle DCA.

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

Recent Articles

  1. Factoring trinomials of the form x^2 + bx + c

    Jul 03, 20 09:51 AM

    factoring trinomials (ax^2 + bx + c ) when a is equal to 1 is the goal of this lesson.

    Read More

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

                                 Follow me on Pinterest

Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.