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Home / Geometry / Prove that when transversal cuts two parallel straight lines, the pair of interior angles on the same side of the transversal are supplementary.

Prove that when transversal cuts two parallel straight lines, the pair of interior angles on the same side of the transversal are supplementary.

Solution:

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Particular enunciation: Let AB||CD and transversal PQ intersects them at E and F respectively. On one side of the transversal PQ the interior angles be ∠BEF and ∠EFD and on the other side the interior angles be ∠AEF and ∠EFC.
Proof: ∠EFD = Corresponding angle ∠PEB.
Now, ∠BEF + ∠EFD = ∠BEF + ∠PEB; [∠EFD = Corresponding angle ∠PEB] = ∠PEF
= 1 straight angle
=2 right angle.
∵ If the sum of two angles is equal to 2 right angles then they are supplementary.
∴ ∠BEF and ∠EFD are supplementary.
Similarly it is shown that ∠AEF and ∠EFC are also supplementary. (Proved)

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