# (a) Find the measure of ∠QPM.

# (b) What are the measure of ∠PQM and ∠PRM?

# (c) If PQ = 6 cm. Find the measure of PR.

**Solution:** Given that, PM⊥QR, ∠QPM = ∠RPM and ∠QPR =90^{0}.

(a) ∠QPM + ∠RPM = ∠QPR

⟹∠QPM + ∠QPM = ∠QPR [∵∠QPM = ∠RPM]

⟹ 2∠QPM = 90^{0} [∵∠QPR =90^{0}]

⟹ ∠QPM = 90^{0}/2

⟹ ∠QPM = 45^{0}

## Therefore the value of ∠QPM is 45^{0}.

(b) Since PM⊥QR, so, ∠PMQ = ∠PMR = 1 right angle = 90^{0}

**In right angle triangle PQM**

∠PQM + ∠QPM = 90^{0}

⟹ ∠PQM + 45^{0} = 90^{0} [From (a)]

⟹ ∠PQM = 90^{0} – 45^{0}

∴ ∠PQM = 45^{0}.

**Similarly, in right angle triangle RPM**

∠RPM + ∠PRM = 90^{0}

⟹ ∠PQM + ∠PRM = 90^{0} [∵∠QPM = ∠RPM]

⟹ 45^{0} + ∠PRM = 90^{0}

⟹ ∠PRM = 90^{0} – 45^{0}

∴ ∠PRM = 45^{0}.

Therefore ∠PQM = 45^{0} and ∠PRM = 45^{0}.

(c) Since ∠PQM = 45^{0} and ∠PRM = 45^{0}

∴ ∠PQM = ∠PRM

⟹ ∠PQR = ∠PRQ

Now in triangle PQR

∠PQR = ∠PRQ

⟹ PR = PQ

⟹ PR = 6 [∵ PQ = 6]

*Therefore PR = 6 cm.*