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Theorem 1 and Theorem 2

Theorem1– Pair Corresponding Angles are Equal

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

1 = 5

2 = ∠ 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: 3 = 5 and 6 = 4
 


Proof: 1 = 3 …. (Vertically opposite angles)
Also
1 = 5 …. (Corresponding angles)
3 = 5 …. (i)
Since ray NM stands on line CD
6 + 5 = 180° …. (Linear pair axiom)
Since ray MN stands on line AB
4 + 3 = 180° …. (Linear pair axiom)
4 + 3 = 6 + 5
But
3 = 5 (Proved in (i))
4= 6
Hence
4 = 6 and 3 = 5.

 



 




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