if PS is the bisector of

1.	if PS is the bisector of
Answer» Given: In figure. PS is the bisector of\xa0{tex}\\angle QPR{/tex} of\xa0{tex}\\triangle PQR{/tex}To prove :{tex}\\frac { Q S } { S R } = \\frac { P Q } { P R }{/tex}Construction: Draw RT \|\| SP to meet QP produced in T.Proof:\xa0{tex}\\because {/tex}\xa0RT\|\|SP and transversal PR intersects them{tex}\\therefore \\angle 1 = \\angle 2{/tex}\xa0(1) ..... Alt. Int.\xa0{tex}\\angle s{/tex}{tex}\\because {/tex}\xa0RT\|\|SP and transversal QT intersects them{tex}\\therefore \\angle 3 = \\angle 4{/tex} (2), ..... corres.\xa0{tex}\\angle s{/tex}But\xa0{tex}\\angle 1 = \\angle 3{/tex}\xa0...... Given{tex}\\therefore \\angle 2 = \\angle 4{/tex}\xa0......From (1) and (2){tex}\\therefore P T = P R{/tex}\xa0(3) ......{tex}\\because {/tex}\xa0Sides opposite to equal angles of a triangle are equal\xa0Now in\xa0{tex}\\Delta \\mathrm { QRT }{/tex}PS = RT .......By construction{tex}\\therefore \\frac { Q S } { S R } = \\frac { P Q } { P T }{/tex}\xa0....... By basic proportionally theorem\xa0{tex}\\Rightarrow \\frac { Q S } { S R } = \\frac { P Q } { P R }{/tex}\xa0......From (3)

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