Question : In $\triangle$PQR, the angle bisector of $\angle$P intersects QR at M. If PQ = PR, then what is the value of $\angle$PMQ?
Option 1: 75°
Option 2: 80°
Option 3: 70°
Option 4: 90°
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Correct Answer: 90°
Solution :
In $\triangle$PQR, PQ = PR
⇒ $\angle$Q = $\angle$R = b
PM is the angle bisector of $\angle$P.
$\angle$QPM = $\angle$RPM = a
Apply angle sum property in $\triangle$PQR,
⇒ $\angle$P + $\angle$Q + $\angle$R = 180°
⇒ $\angle$P + b + b = 180°
⇒ $\angle$P = 180° – 2b
$\angle$QPM = $\frac{180° – 2b}{2}$ = 90° – b
In $\triangle$PQM,
Let $\angle$PMQ = $\theta$
⇒ $\angle$QPM + $\angle$Q + $\angle$M = 180°
⇒ 90° – b + b + $\theta$ = 180°
⇒ $\theta$ = 90°
Hence, the correct answer is 90°.
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