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 pre-image and the image are parallel, since the distances between the pre-image 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).

  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 non-adjacent interior angle.

  5.

5. Congruent angles are angles of equal measure.

6. Contradiction steps 4 and 5.

  

1. Given

  2.

2. Assumption leading to a contradiction.

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. Contradiction steps 3 and 5.