Final answer:
To prove that interior angles 1 and 3 are congruent with parallel lines and a transversal, we use the concept of alternate interior angles, which are equal in measure. Therefore, if angle 1 equals angle 2 and angle 3 equals angle 2, angle 1 must also be congruent to angle 3.
Step-by-step explanation:
To prove that interior angles 1 and 3 are congruent when the lines are parallel with a transversal, we can use the concept of alternate interior angles. When two lines are parallel and cut by a transversal, the alternate interior angles are equal in measure.
In the case of angle 1 and angle 3, since the lines are parallel by assumption, angle 1 is congruent to angle 2 (being alternate interior angles with line as the transversal). Similarly, angle 3 is congruent to angle 2 (being alternate interior angles with line as the transversal). Therefore, if angle 1 is congruent to angle 2 and angle 3 is congruent to angle 2, by the Transitive Property of Equality, angle 1 is congruent to angle 3.
This result is essential in geometry and underlies many proofs and theorems.