Question

If the angle bisector of angle PQR and angle PRQ of triangle PQR meet at a point M, then prove that angle QMR=90°+1/2of angle P

Answer 1

Given that

OQ and OR are angle bisector of Q and R

PQR is a triangle

also from figure OQR is a triangle

Let angle Q= 2x

angle R = 2y

Now in ∆ PQR

angle Q + angle R + angle P = 180°

or 2x + 2y +angle P = 180°

=> x+y = 90° - 1/2 angle P .......... (i)

Now in ∆OQR

angle RQO + angle QRO + angle O = 180

as angle RQO = x (from figure)

and angle QRO = y(from figure)

=> x + y + angle O = 180°

=> angle O = 180° - (x+y) ........(ii)

Now from (i)and (ii)

Angle O = 180° - ( 90° - 1/2 angle P )

=> angle O = 90° + 1/2 angle P

Answer 2

refer pic above plz....