If two lines are cut by a transversal so that the exterior angles on the same side of the transversal are supplementary, then these lines are parallel.

Given:
Lines n and m; transversal t
 

Prove : m||n

Proof:


1: - Given

2: - If the exterior sides of two adjacent angles form a straight line, these angles are supplementary

3: - From 1 and 2

m||n - If two lines (m, n) are cut by a transversal (t) so that the corresponding angles (1, 2) are congruent, then these lines are parallel

If two lines are cut by a transversal so that the interior angles on the same side of the transversal are supplementary, then these lines are parallel

Given: Lines n and m; transversal t



Prove:
m||n





Proof:

- Given
- If the exterior sides of two adjacent angles form a straight line, these angles are supplementary

- If two angles are supplementary  to the same angle , they are congruent

m||n - If two lines (m, n) are cut by a transversal (t) so that the corresponding angles (1, 2) are congruent, then these lines are parallel

If two lines are cut by a transversal so that the alternate interior angles are congruent, then these lines are parallel.

Given:
Lines n and m and transversal t


 PROVE: n||m

 PROOF:

 - Given

 -  If two lines intersect, then the vertical angles formed are congruent.

 - Transitive Property of Congruence

m||n - If two lines (m, n) are cut by a transversal (t) so that the corresponding angles () are congruent, then these lines are parallel

If two lines are cut by a transversal so that the corresponding angles are congruent, then these lines are parallel

n and m cut by transversal t

GIVEN: n and m cut by transversal t

 PROVE: n||m

 PROOF:

Suppose that n and m are not parallel. Then a line r can be drawn through point P that is
parallel to m; (r||m)

1:  - angles correspond r||m , transversal t

2:  - Given

3: -  From 2,3

4: - Definition of r

5: - From 3,4

Line r paraller to m

6: Then r and n must coincide, and it follows that n||m