Question

the bisectors of angle P and angle Q of a quadrilateral PQRS meet at a point M. if angle R equal to 95 degree and Angle S equal to 31 degree find the measure of angle PMQ​

Answer 1

Answer:

W.K.T , The sum of interior angles of quadrilateral is 360 degree, so we have;

∠P+∠Q+∠R+∠S = 360°

∠P+∠Q+100°+50° = 360°

∠P+∠Q = 360°−150°

∠P+∠Q = 210°......................(i)

Now, OP and OQ are the bisectors of angles P and Q respectively.

So, ∠P = 2∠OPQ and ∠Q = 2∠OQP

Equation (i) becomes

2∠OPQ+2∠OQP = 210°

2(∠OPQ+∠OQP) = 210°

∠OPQ+∠OQP = 105°..................(ii)

In triangle POQ,

∠POQ+∠OPQ+∠OQP = 180°

∠POQ+105° = 180° [Using(ii)]

∠POQ = 180°−105°

= 75°

Therefore, angle POQ is 75 degree.