Question

PLEASE HELP
(look at pictures)

Answer 1

A two-column proof to prove that lines p and q are parallel lines should be completed as follows;

Statement Reason______________

1. ∠1 ≅ ∠3 1. Given

2. ∠1 and ∠3 are vertical ∠s 2. Definition of vertical angles

3. ∠1 ≅ ∠3 3. Vertical angles theorem

4. ∠2 ≅ ∠3 4. Transitive property

5. p ║ q 5. Converse of corresponding angles theorem

In Mathematics and Euclidean Geometry, the vertical angles theorem states that two opposite vertical angles that are formed whenever two lines intersect each other are always congruent, which means being equal to each other.

The converse of corresponding angles theorem is a theorem which states that corresponding angles are always congruent when the transversal intersects two parallel lines.

In this context, we can logically deduce that lines p and q are parallel lines based on the wo-column proof above.

Answer 2

This is a theorem called Converse of Alternate Exterior Angles that states that if two lines are cut by a transversal and the alternate exterior angles are congruent, then the lines are parallel. Moreover, this theorem is based upon the corresponding Angles Converse Postulate that states that if two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel. We don't need to prove this postulate, it's assume to be true. So our goal is to get corresponding angles congruent in order to use the corresponding Angles Converse Postulate,

1.

Reason: Given

Statement:
`\angle1\cong\angle2`

2.

Reason: Def of vertical
`\angle s</p><p>`

Statement:
`\angle1\:and\:\angle3 \ are \ vertical`

3.

Reason: Def of vertical
`\angle s`

Statement:
`\angle1\cong\angle3`

4.

Reason: Transitive Property

Statement:
`\angle2\cong\angle3`

5.

Reason: corresponding Angles Converse Postulate

Statement:
`p\parallel q`