PN 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)
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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)
∴, ∠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)
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