Question

The bisector of any two adjacent angles of a rhombus form a right-angled triangle.
State true or false.

• A. True
• B. False

Answer 1

The correct option is A True

Let ABCD be a rhombus with AB and DC be 1 pair of opposite parallel sides and AD and BC be the other pair of opposite parallel sides.

(1).Now consider parallel sides $AB$ and $DC$.

(2).The side $AD$ is the intersection with these parallel sides.

(3) angle A and angle D are the internal angles on the same side of the parallel lines $AB,DC$ formed with the intersection AD.

Therefore angle $\angle A+\angle D=180°.$

(4). Dividing both sides by 2, $\left(1/2\right)\angle A+\left(1/2\right)\angle D=\left(1/2\right)x180=90°.$

(5). Let bisectors of angle A and angle D meet at O.

Then in triangle AOD ,$\left(1/2\right)\angle A+\left(1/2\right)\angle +\angle AOD=180°$

Hence angle $AOD=180°-\left(\right(1/2)\angle A+(1/2)\angle D$

$=180°-90°=90°$

Answer = 1