Question

In the given figure, ON and UL are respectively the bisectors of angle P and angle U of a quadrilateral JUMP meet MU and JP produced in N and L.Prove that angle JPM + angle JUM = 2 (angle N + angle L)

Answer 1

PN is the bisector of ∠P.
∴, ∠JPN=∠NPM ------------------(1)
UL is the bisector of ∠U.
∴, ∠JUL=∠LUM -------------------(2)
∴, ∠JPM+∠JUM
=(∠JPN+∠NPM)+(∠JUL+∠LUM)
=2∠JPN+2∠LUM  [using (1) and (2)]
=2∠PNM+2∠JLU
[∵, from the given figure, JL||NM, PN traversal
∴, ∠JPN=∠PNM (alternate angles) and 
∵, JL||NM, UL traversal
∴, ∠JLU=∠LUM (alternate angles)]
=2(∠N+∠L) (Proved)