Step-by-step explanation:
Given that PQ ║RS, cut by a transversal, t.
In order to prove that m ∠ 3 = m ∠ 5, we must first define the necessary terms related to the given problem.
Definitions:
Vertical angles are formed by two intersecting lines, whose sides form two pairs of opposite rays.
- The Vertical Angles Congruence Theorem states that vertical angles have measures that are congruent.
Two angles are considered as corresponding angles if they are located on corresponding positions. These angles have the same measure.
- The Corresponding Angles Theorem states that if two parallel lines intersect a transversal, then the measures of the corresponding angle pairs are congruent.
The following are the corresponding angles in the given diagram are:
∠ 1 and ∠5
∠2 and ∠6
∠3 and ∠7
∠4 and ∠8
Alternate interior angles are in between two parallel lines that lie on the opposite sides of the transversal, and do not share a common vertex. Alternate interior angles have the same measure.
- According to the Alternate Interior Angles Theorem, if two parallel lines intersect a transversal, then the measures of the alternate interior angle pairs are congruent.
The alternate interior angles in the given diagram are:
∠3 and ∠5
∠4 and ∠6
- The Transitive Property of Congruence states that if ∠A ≅ ∠B, and ∠B ≅ ∠C, then ∠A ≅∠C.
Prove that m ∠3 = m ∠5:
1.) Statement: PQ ║RS.
Reason: Given.
2.) Statement: ∠3 ≅ ∠7
Reason: Corresponding Angles Theorem.
3.) Statement: ∠7 ≅ ∠5
Reason: Vertical Angles Congruence Theorem.
4.) Statement: ∠3 ≅ ∠5
Reason: Transitive Property of Congruence.
