"Angle Q is greater than angle R and M is a point on QR such that PM is the bisector - Maths - Lines and Angles

Question

"Angle Q is greater than angle R and M is a point on QR such
that PM is the bisector of angle QPR If the perpendicular from P on
QR meets at N, prove that angle MPN =1/2(angle Q-angle R)

S. Usha Rani · Accepted Answer

<img alt="" width="320" height="123" src=
"https://s3mn%20mnimgs%20com/img/shared/discuss_editlive/2198141/2013_08_27_15_09_26/image1476527584997206675%20jpg">
âˆ MPN =
âˆ P-âˆ QPN-âˆ MPR
=âˆ P-âˆ QPN-1/2
âˆ P(Because MP is the angular bisector of
âˆ P)
=1/2 âˆ P -âˆ QPN
=1/2 âˆ P -(90 - âˆ Q) (because
triangle PQN is right angled)
= 1/2 (180-âˆ Q-âˆ R) -
(90-âˆ Q) (because the sum of angles in triangle
PQR is 180)
=90 - âˆ Q/2 - âˆ R/2 -90
+âˆ Q
= âˆ Q/2-âˆ R/2
=1/2(âˆ Q-âˆ R)

"Angle Q is greater than angle R and M is a point on QR such that PM is the bisector of angle QPR If the perpendicular from P on QR meets at N, prove that angle MPN =1/2(angle Q-angle R)