In the given figure, seg PT is the bisector of $∠$ QPR. A line through R intersects ray QP at point S. Prove that PS = PR

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Solution

According to the figure, we have
PT || SR
If PT || SR and PS is a transversal line, then
$∠$ QPT = $∠$ PSR (Corresponding angles) ....(1)
If PT || SR and PR is a transversal line, then
$∠$ TPR = $∠$ PRS (Alternate angles) ....(2)
We are given that seg PT is the bisector of $∠$ QPR.
∴ $∠$ QPT = $∠$ TPR ...(3)
From (1), (2) and (3)
$∠$ PSR = $∠$ PRS
The sides opposite to equal angles are also equal in a triangle.
∴ PS = PR

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In the given figure, seg PT is the bisector of ∠QPR. A line through R intersects ray QP at point S. Prove that PS = PR

In the given figure, seg PT is the bisector of $∠$ QPR. A line through R intersects ray QP at point S. Prove that PS = PR