Question

diagonals of a rectangle are congruent

Answer 1

Given a rectangle, prove that the diagonals are congruent.

Given: Rectangle ABCD

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

AB ≅ segment DC

∠ABC ≅ ∠DCB

BC ≅ BC

Therefore, by SAS, triangle ABC ≅ triangle DCB.

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

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Answer 2

Answer:

The proof is given below.

Step-by-step explanation:

Here we have been asked to prove that the diagonals of the rectangle are equal for this we consider a rectangle MNOQ as attached below. Here we have to prove that MO and QN are equal diagonals.

Here we have MN equal to OQ since the opposite sides in the respective rectangle are equal in length. And we also know that a rectangle has an angle of 90° on all of its corresponding four vertices.

Using these we have in Δ MNO and ΔQON.

MN = QO (Opposite sides in a rectangle are equal)

∠ MNO = ∠ QON = 90° (since the angles in the rectangles are right angles)

NO = ON ( common in both)

Therefore Δ MNO ≅ ΔQON by SAS congruency criterion.

Hence MO = QN by C.P.C.T

Therefore the diagonals of the rectangle are congruent hence proved.