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प्रश्न
In the given figure; ABC, AEQ and CEP are straight lines. Show that ∠APE and ∠CQE are supplementary.
उत्तर
Given – In the figure, ABC, AEQ and CEP are straight lines
To prove – ∠APE + ∠CQE = 180°
Construction – Join EB
Proof – In cyclic quad ABEP,
∠APE + ∠ABE = 180° ...(1)
Similarly, in cyclic quad BCQE
∠CQE + ∠CBE = 180° ...(2)
Adding (1) and (2),
∠APE + ∠ABE + ∠CQE + ∠CBE = 180° + 180° = 360°
`=>` ∠APE + ∠CQE + ∠ABE + ∠CBE = 360°
But, ∠ABE + ∠CBE = 180° ...[Linear pair]
∴ ∠APE + ∠CQE + 180° = 360°
`=>` ∠APE + ∠CQE = 360° – 180° = 180°
Hence ∠APE and ∠CQE are supplementary.
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