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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:

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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