# Q: 7  ABCD is a rhombus. Show that diagonal AC bisects $\small \angle A$ as well as $\small \angle C$ and diagonal BD bisects $\small \angle B$ as well as $\small \angle D$.

In $\triangle$ ADC,

AD = CD          (ABCD is a rhombus)

$\angle$3=$\angle$1.................1(angles opposite to equal sides are equal )

$\angle$3=$\angle$2.................2 (alternate angles)

From 1 and 2, we have

$\angle$1=$\angle$2.................3

and        $\angle$1=$\angle$4.................4 (alternate angles)

From 1 and 4, we get

$\angle$3=$\angle$4.................5

Hence, from 3 and 5, diagonal AC bisects angles A as well as angle C.

In $\triangle$ ADB,

AD = AB          (ABCD is a rhombus)

$\angle$5=$\angle$7.................6(angles opposite to equal sides are equal )

$\angle$7=$\angle$6.................7 (alternate angles)

From 6 and 7, we have

$\angle$5=$\angle$6.................8

and        $\angle$5=$\angle$8.................9(alternate angles)

From 6 and 9, we get

$\angle$7=$\angle$8.................10

Hence, from 8 and 10, diagonal BD bisects angles B as well as angle D.

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