Question

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

Answer 1

Given :   bisectors of angles ∠PQR and ∠PRQ of triangle PQR meet at a point M

To find :  prove that angle QMR = 90+1/2 angle P

Solution:

in Δ PQR

∠PQR + ∠PRQ  + ∠P  = 180°   ( sum of angles of triangle)

=>∠PQR + ∠PRQ = 180°  - ∠P

in Δ MQR

∠MQR + ∠MRQ  + ∠QMR  = 180°   ( sum of angles of triangle)

∠MQR  = ∠PQR/2

∠MRQ =  ∠PRQ/2

=> ∠PQR/2 + ∠PRQ/2 + ∠QMR  = 180°

=> (∠PQR + ∠PRQ )/2 + ∠QMR  = 180°

=> (180°  - ∠P)/2 + ∠QMR  = 180°

=> 90° - ∠P/2 +  ∠QMR  = 180°

=>  ∠QMR  = 90° + ∠P/2

QED

Hence proved

Learn More:

The interior angle bisector of angle B, and the exterior bisector of ...

https://brainly.in/question/13087951

The bisectors of the angles A, B of ∆ABC intersect in I, the bisectors ...

https://brainly.in/question/14495363