Diagonals of Rectangle are Equal

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Theorem

The diagonals of a rectangle are equal.


Proof

Diagonals of Rectangle are Equal.png

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 Equality it follows that $\triangle ADC = \triangle DAB$.

Thus $AC = BD$.

$\blacksquare$