Answer:
12. ∠2 ≅ ∠3
The parallel Lines for these angles are q and p.
The theorem that justifies this is Corresponding Angles, because both are in the same position (the upper right-hand corner) in their group of 4 angles. These angles are also Congruent, because the transversal intersects two parallel lines and two corresponding angles are congruent. Q and P would be the parallel lines and R would be the transversal.
13. ∠6 ≅ ∠7
The parallel lines for these angles would be Q and P.
The theorem that justifies this is Alternate Interior Angles, because this pair of angles are on opposite sides of the Transversal line and on the inside of the parallel lines. This also makes them Congruent angles, because the transversal intersects two parallel lines and two Alternate Interior Angles are Congruent. The two parallel lines are Q and P are parallel because line S intersect them, making line S the Transversal.
14. ∠1 ≅ ∠4
The parallel lines for these angles are R and S.
The theorem that justifies this is Alternate Exterior Angles, because angles 1 and 4 are on opposite sides of the transversal as well as the outside of the parallel lines. This also makes these two angles Vertical Angles, because when a transversal intersects two parallel lines, and the angles are across from each other on opposite sides of the transversal, they are vertical angles. Vertical Angles are always Congruent. So R and S are the parallel lines and line Q is the Transversal because it intersects R and S.
15. m∠5 + m∠8 = 180°
The parallel lines for angles 5 and 8 are R and S.
The theorem that justifies this is Same-Side Interior Angles, because they are on the same side of the Transversal and on the inside of the parallel lines. This also makes angles 5 and 8 Supplementary, because a transversal intersects two parallel lines and Same-side Interior Angles are Supplementary. The parallel lines are R and S because Transversal line P intersects them.
Step-by-step explanation:
Explanations are within the answers for each question above.
Hope this helps!! :)
