Proof:
m∠2 + m∠3 = 180° (Given)
∠2 and 5 are supplementary (Given)
m∠2 + m∠5 = 180° (Substitution)
∠3 and ∠5 are supplementary (Linear Pair Postulate)
p || q (Alternate Interior Angles Theorem)
Solution to Proof
Given:
m∠2 + m∠3 = 180°
∠2 and ∠5 are supplementary
Prove:
p || q
Proof:
m∠2 + m∠3 = 180° (Given)
∠2 and 5 are supplementary (Given)
∴ m∠2 + m∠5 = 180° (Substitution)
∴ ∠3 and ∠5 are supplementary (Linear Pair Postulate)
∴ p || q (Alternate Interior Angles Theorem)
Conclusion:
Therefore, if m∠2 + m∠3 = 180° and ∠2 and ∠5 are supplementary, then p || q.
Image:
The image provided shows a diagram of the situation described in the proof. The lines p and q are parallel, and the angles ∠2, ∠3, and ∠5 are supplementary.
The Linear Pair Postulate states that two angles that are supplementary form a straight line. The Alternate Interior Angles Theorem states that if two lines are parallel and cut by a transversal, then the alternate interior angles are congruent.
In the proof, we first use the given information to show that ∠3 and ∠5 are supplementary. Then, we use the Alternate Interior Angles Theorem to show that p || q.