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Theorem 1: Equal Chords Subtend Equal Angles at the Center

Seg PM and ...

Question

Seg $P M$ and Seg $P N$ are congruent chords of a circle with centre $C$ . Show that the ray $P C$ is the bisector of $∠ N P M$

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Solution

In $Δ C P M$ and $Δ C P N$ ,

$C P = C P$

$C M = C N$ (radius)

$M P = N P$ (congruent chords)

$\Rightarrow$ By S.S.S congruency,

$Δ C P M ≅ Δ C P N$

$\Rightarrow ∠ M P C = ∠ N P C$

$\Rightarrow ∠ x = ∠ y$ .

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