Diagonals of Rectangle are Equal

Theorem

The diagonals of a rectangle are equal.

Proof

Let $ABCD$ be a rectangle.

The diagonals of $ABCD$ are $AC$ and $BD$.

Then $\angle ADC = \angle DAB$ as both are right angles by definition of rectangle.

By Rectangle is Parallelogram, $ABCD$ is also a type of parallelogram.

Thus by Opposite Sides and Angles of Parallelogram are Equal $AB = DC$.

Thus we have:

$AB = DC$

$\angle ADC = \angle DAB$

$AD$ is common to both $\triangle ADC$ and $\triangle DAB$

and so by Triangle Side-Angle-Side Congruence:

$\triangle ADC = \triangle DAB$

Thus:

$AC = BD$

$\blacksquare$