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In the adjoining PQR\triangle PQR, ray PA and ray QB are the bisectors of P\angle P and Q\angle Q respectively. If R=70\angle R=70^\circ. Find mPMQm \angle PMQ.

Question Illustration

Options

Option A

120°

Option B

130°

Option C is correct

125°

Option D

135°

Explanation

In PQR\triangle PQR, the sum of angles is 180180^\circ. Given R=70\angle R=70^\circ, we have P+Q=18070=110\angle P + \angle Q = 180^\circ - 70^\circ = 110^\circ. Since PA and QB are angle bisectors, in PMQ\triangle PMQ, QPM=12P\angle QPM = \frac{1}{2}\angle P and PQM=12Q\angle PQM = \frac{1}{2}\angle Q. The sum of angles in PMQ\triangle PMQ is 180180^\circ, so PMQ+QPM+PQM=180\angle PMQ + \angle QPM + \angle PQM = 180^\circ. This simplifies to PMQ+12(P+Q)=180\angle PMQ + \frac{1}{2}(\angle P + \angle Q) = 180^\circ. Substituting the sum of P\angle P and Q\angle Q, we get PMQ+12(110)=180\angle PMQ + \frac{1}{2}(110^\circ) = 180^\circ. Thus, PMQ+55=180\angle PMQ + 55^\circ = 180^\circ, which means PMQ=18055=125\angle PMQ = 180^\circ - 55^\circ = 125^\circ. Alternatively, the angle formed by two angle bisectors is given by 90+12R=90+12(70)=90+35=12590^\circ + \frac{1}{2}\angle R = 90^\circ + \frac{1}{2}(70^\circ) = 90^\circ + 35^\circ = 125^\circ.