Maharashtra State BoardSSC (English Medium) 8th Standard
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Properties of Angles Formed by Two Parallel Lines and a Transversal - Axiom: If a Transversal Intersects Two Parallel Lines, Then Each Pair of Alternate Interior Angles Are Equal.

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If a Transversal Intersects Two Parallel Lines, Then Each Pair of Alternate Interior Angles Are Equal.

Given: Two parallel lines AB and CD.

Let PS be the transversal intersecting AB at Q and CD at R.

To Prove: Each pair of alternate interior angles are equal.

i.e., ∠BQR = ∠ CRQ

and ∠ AQR = ∠ QRD.

Proof: 

First, we will prove ∠ BQR = ∠ CRQ.

For lines AB & CD, with transversal PS.

∠ AQP = ∠ CRQ    .....(Corresponding angles)(1)

For lines AB & PS,

∠ AQP = ∠ BQR    ......(Vertically opposite angles)(2)

From (1) and (2),

∠ BQR = ∠ CRQ

Similarly, we can prove

∠ AQR = ∠ QRD

Hence, Pair of alternate interior angles are equal.

Hence proved.

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