# Theorem 1 and Theorem 2

Theorem1â€“ Pair Corresponding Angles are Equal

**Axiom:** If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.

âˆ 1 = âˆ 5

âˆ 2 = âˆ 6 are the examples of equal corresponding angles.

**Converse :** If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.

Theorem 2

If a transversal intersects two parallel lines, then each pair of alternate interior angles are equal.

**Given : ** Two lines AB and CD such that AB | | CD and a transversal EF intersects AB at M and CD at N.

To Prove: âˆ 3 = âˆ 5 and âˆ 6 = âˆ 4

**Proof:**** ** âˆ 1 = âˆ 3 â€¦. (Vertically opposite angles)

Also âˆ 1 = âˆ 5 â€¦. (Corresponding angles)

âˆ´ âˆ 3 = âˆ 5 â€¦. (i)

Since ray NM stands on line CD

âˆ´ âˆ 6 + âˆ 5 = 180Â° â€¦. (Linear pair axiom)

Since ray MN stands on line AB

âˆ´ âˆ 4 + âˆ 3 = 180Â° â€¦. (Linear pair axiom)

âˆ´ âˆ 4 + âˆ 3 = âˆ 6 + âˆ 5

But âˆ 3 = âˆ 5 (Proved in (i))

âˆ´ âˆ 4= âˆ 6

Hence âˆ 4 = âˆ 6 and âˆ 3 = âˆ 5.