Question

Given that lines a and b are parallel, angles 1 and 5 are congruent because they are corresponding angles, and angles 1 and 4 are congruent because they are vertical angles. Then, by which property are angles 4 and 5 congruent? Together, these statements prove which theorem? A) Transitive Property of Congruency; If parallel lines have a transversal, then supplemental angles are congruent. B) Transitive Property of Congruency;If parallel lines have a transversal, then alternate interior angles are congruent. C) Symmetric Property of Congruency; If parallel lines have a transversal, then alternate exterior angles are congruent. D) Commutative Property of Congruency; If parallel lines have a transversal, then same side interior angles are congruent.

Answer 1

Option B)

Explanation:

Given :

Given that lines a and b are parallel, angles 1 and 5 are congruent because they are corresponding angles, and angles 1 and 4 are congruent because they are vertical angles

To find : by which property are angles 4 and 5 congruent

Solution :

We know that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Also, we know that if two things are equal to the same thing then they are equal to each other . In this case, we can say that if two angles are congruent to a third angle, then they are congruent to each other. As angles 4 and 5 are both congruent to angle 1, they are congruent to each other but angles 4 and 5 are alternate interior angles. So, if parallel lines have a transversal, alternate interior angles are congruent.

Answer 2

Transitive Property of Congruency;If parallel lines have a transversal, then alternate interior angles are congruent.

The Transitive Property of Congruency:

If 2 angles are congruent to a third angle, then they are congruent to each other. So, since angles 4 and 5 are both congruent to angle 1, they are congruent to each other.

Angles 4 and 5 are alternate interior angles. Therefore, if parallel lines have a transversal, alternate interior angles are congruent.