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2 geometry problems, angle related (1st is a classic, 2nd angle bisectors)

Source: Gulf Mathematical Olympiad 2015 p2

August 23, 2019
geometryanglesangle bisectorAngle Chasing

Problem Statement

a) Let UVWUVW , U′V′W′U'V'W' be two triangles such that VW=V′W′,UV=U′V′,∠WUV=∠W′U′V′. VW = V'W' , UV = U'V' , \angle WUV = \angle W'U'V'. Prove that the angles ∠VWU,∠V′W′U′\angle VWU , \angle V'W'U' are equal or supplementary.
b) ABCABC is a triangle where ∠A\angle A is obtuse. take a point PP inside the triangle , and extend AP,BP,CPAP,BP,CP to meet the sides BC,CA,ABBC,CA,AB in K,L,MK,L,M respectively. Suppose that PL=PM.PL = PM . 1) If APAP bisects ∠A\angle A , then prove that AB=ACAB = AC . 2) Find the angles of the triangle ABCABC if you know that AK,BL,CMAK,BL,CM are angle bisectors of the triangle ABCABC and that 2AK=BL2AK = BL.
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