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Quadrilaterals — Maths STD 9 — Question

Gujarat BoardEnglish MediumSTD 9MathsQuadrilaterals5 Marks
Question
$A B C D$ is a rectangle in which diagonal $A C$ bisects $\angle A$ as well as $\angle C$. Show that:
i. $A B C D$ is a square
ii. diagonal $B D$ bisects $\angle B$ as well as $\angle D$.
✓

Answer



Given : $ABCD$ is a rectangle,
$\angle A = \angle C$
$\frac{1}{2} \angle A=\frac{1}{2} \angle C$
To prove: $ABCD$ is a square proof:
i. $\angle DAC =\angle DCA$ ($AC$ bisects $A$ and $C)$
$C D=D A$ (Sides opposite to equal angles are also equal)
However,
$D A=B C$ and $A B=C D$ (Opposite sides of a rectangle are equal)
$AB=BC=CD=DA$
$A B C D$ is a rectangle and all of its sides are equal.
Hence, $A B C D$ is a square
ii. Let us join $BD$
In $\triangle B C D$,
$B C=C D$ (Sides of a square are equal to each other)
$\angle C D B=\angle C B D$ (Angles opposite to equal sides are equal)
However,
$\angle C D B=\angle A B D$ (Alternating interior angles for $A B \| C D$ )
$\angle C B D=\angle A B D$
$B D$ bisects $\angle B$
Also,
$\angle CDB=\angle ABD$
$BD$ bisects $\angle D$.

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