
The following theorems are demonstrations of proving parallel theorems true.
More than one method of proof exists for each of the these theorems.
On this page, only one style of proof will be provided for each theorem.
Prove Corresponding Angles Congruent: (Transformational Proof)
If two parallel lines are cut by a transversal, the corresponding angles are congruent.

1. Let's look first at ∠BEF.
Using
as the translation vector, slide ∠BEF down the ray such that its vertex remains on the ray and will coincide with point F.
Point F will become the image of point E. 

2. We now know that When studying translations, we discovered that the corresponding sides (lines) of the preimage and the image are parallel, since the distances between the preimage points and image points are constant (equal).


3. Now, is ? Yes!
In a plane, if two lines are parallel to the same line, they will be parallel to each other.

4. Does actually coincide with ? Yes!
Since these segments are parallel and share a common end point, F ( E'), they must be on the same line. The Parallel Postulate states that through any point ( F ) not on a given line ( ), only one line may be drawn parallel to the given line. 

5. We now know that ∠1 ∠2. Being a rigid transformation, translations preserve angle measure, making m∠1 = m∠2 (or congruent). 

Prove Lines are Parallel: (Indirect Proof)
If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.


1. 
1. Given 

2. Assumption leading to a contradiction. 

3. In a plane, two distinct lines are either parallel or intersecting. 
4. 
4. The measure of an exterior angle of a triangle is greater than the measure of either nonadjacent interior angle. 
5. 
5. Congruent angles are angles of equal measure. 

6. Contradiction steps 4 and 5. 

Prove Alternate Interior Angles Congruent: (Indirect Proof)
If two parallel lines are cut by a transversal, the alternate interior angles are congruent.



1. Given 
2. 
2. Assumption leading to a contradiction. 
3. 
3. An angle may be copied. 

4. If 2 lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel. 

5. Parallel Postulate: Through a point not on a given line, only one line can be drawn parallel to the given line. 
6. 
6. Contradiction steps 3 and 5. 

NOTE: The reposting of materials (in part or whole) from this site to the Internet
is copyright violation
and is not considered "fair use" for educators. Please read the "Terms of Use". 
