Question

In the given figure, PS=PT and S and T are points on QR such that QS=TR. Show that PQ=PR​

Answer 1

Answer:

In a triangle PQR, PT is perpendiclar to QR, PS is the bisector of angle P. How can you show that angle TPS = (1/2) * (angle Q - angle R)?

Given

△PQR , PS is bisector of ∠P and PT is perpendicular to QR .

To prove

∠TPS=12(∠Q−∠R)

Proof

In △TPS ,

∠TPS+∠PTS+∠ PST=180°

∠TPS+90°+(∠SPR+∠SRP)=180°

(PT is perpendicular to QR and angle PST is exterior angle of triangle PRS)

∠TPS+90°+12∠P+∠R=180°

(PS is bisector)

∠TPS+90°+12(180°−∠Q−∠R)+∠R=180° (Angle sum property of triangle)

∠TPS+90°+90°−12∠Q−12∠R+∠R=180°

∠TPS+180°−12∠Q+12∠R=180°

∠TPS=180°−180°+12∠Q−12∠R

∠TPS=12(∠Q−∠R)