Question

Ray $PQ$ and ray $PR$ are perpendicular to each other. Points $B$ and $A$ are in the interior and exterior of $\angle QPR$ respectively. Ray $PB$ and ray $PA$ are perpendicular to each other. Draw a figure showing all these rays and write

1. A pair of complementary angles

2. A pair of supplementary angles

3.A pair of congruent angles.

Answer 1

Ray $PQ$ and $PR$are perpendicular each other. Point $B$ is the interior and Point $A$ is the exterior of $\angle QPR$.

Ray $PB$ and $PA$ are perpendicular to each other.

$\left(i\right)$ A pair of Complementary angles:

$\Rightarrow$ $\angle QPB$ and $\angle BPR$ because their sum is ${90}^o$

$\Rightarrow$ $\angle BPR$ and $\angle RPA$ because their sum is ${90}^o$

$\left(ii\right)$ A pair of Supplementary angles:

$\Rightarrow$ $\angle QPR$ and $\angle BPA$ because their sum is ${180}^o$

$\left(iii\right)$ A pair of congruent angles:

$\Rightarrow$ $\angle BPA\equiv \angle QPR$ [ Each angle is ${90}^o$ ]

$\Rightarrow$ $\angle BPA=\angle QPR$

So, $\angle RPA+\angle BPR=\angle BPR+\angle QPB$ [ Using angle addition property ]

So, $\angle RPA=\angle QPB$

Hence, $\angle RPA\equiv \angle QPB$