Chapter 9.2, Problem 5CE

To determine

To find:The congruent angles in given diagram as well as the pairs of complementary angles. Also to find the pair of supplementary angles.

Answer to Problem 5CE

(i) Congruent angles pair:

(i) $\angle APO,\angle BPO$ (ii) $\angle POB,\angle POA$ (iii) $\angle PBO,\angle PAO$

(ii) Complementary angles pair: (i) $\angle POA\text{and }\angle APO$ (ii) $\angle POB\text{ and }\angle BPO$ .

(iii) Supplementary angles pair: $\angle PBO,\angle PAO$

Explanation of Solution

Giveninformation: The below diagram is given,

  

Concept used:Following properties are used here,

1. In a circle, any line from center of circle to the point of contact of tangent is always perpendicular to tangent.
2. The line passing from the center and intersecting point of two tangents to its circle, always bisect the two angles, forming on the center and intersecting point.
3. In a right triangle, remaining two angles, except right angle, are complementary angles

Also, if the sum of two angles is $90°$ , so these are complementary and if their sum is $180°$ , so these angles are supplementary.

Calculation:So, in given figure as $OA\perp AP$ , so $\angle PAO={90}^0$ and thus $\Delta APO$ is a right triangle. Similarly as $OB\perp BP$ , so $\angle PBO={90}^0$ . Hence, $\Delta BPO$ is a right triangle. Thus, using first property, $\angle PBO=\angle PAO={90}^0$ . So, these two angles are congruent.

Also, the sum of these two angles will be 180 degree, so these two angles are supplementary also.

Further, in right triangle $\Delta APO$ , as angle A is 90 degree, so by angle sum property of triangle,

  $\angle POA+\angle APO={90}^0$ , hence these two angles are complementary to each other. Similarly in right triangle $\Delta BPO$ , $\angle POB+\angle BPO={90}^0$ . So, these two are also complementary angles. Also, using third property, another sets of congruent angles are $\angle POB=\angle POA$ and $\angle APO=\angle BPO$ .

Conclusion: Thus, it is concluded that pairs of congruent angles are

(i) $\angle APO,\angle BPO$ (ii) $\angle POB,\angle POA$ (iii) $\angle PBO,\angle PAO$ .

Pairs of complementary angles are

(i) $\angle POA\text{and }\angle APO$ ii) $\angle POB\text{ and }\angle BPO$ .

Further only pair of supplementary angles is $\angle PBO,\angle PAO$ .