Diagonals of Rectangle are Equal
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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:
- $AC = BD$
$\blacksquare$