Question

1. determine which lines are parallel based on the diagram below.
2. given: ∠a = ∠5. prove: m||n

statements reasons

1. ∠a=∠5 1. given
2. ∠1=∠a 2.
3. ∠1=∠5 3. transitive property
4. m||n 4.

answer choices: vertical angle theorem, corresponding angle converse, vertical angle converse, alternate exterior angle theorem

1. identify the three correct angles that are congruent to ∠1

Explanation:

4.

Step1: Recall parallel - line angle relationships

When two parallel lines are cut by a transversal, corresponding angles are equal, alternate - interior angles are equal, and alternate - exterior angles are equal.
We know that if two lines are parallel, then the sum of same - side interior angles is 180°.
For line \(a\) and \(c\), the angles formed by the transversal are \(115^{\circ}\) and \(65^{\circ}\) (since \(180 - 115=65\)). Since \(a\) and \(c\) have corresponding angles that are equal in measure (not shown in the given angle - measure way for this pair directly, but we can consider the supplementary and corresponding relationships), we check other pairs.
For line \(b\) and \(c\), the angles formed by the transversal are \(115^{\circ}\) and \(115^{\circ}\). Since these are corresponding angles and are equal, \(b\parallel c\).

Step1: Justify \(\angle1=\angle4\)

\(\angle1\) and \(\angle4\) are vertical angles. By the Vertical Angle Theorem, vertical angles are congruent. So the reason for \(\angle1 = \angle4\) is the Vertical Angle Theorem.

Step2: Justify \(m\parallel n\)

Since \(\angle1=\angle5\) (by the transitive property from \(\angle A=\angle4\) and \(\angle A = \angle5\)), and \(\angle1\) and \(\angle5\) are corresponding angles. By the Corresponding Angle Converse (if corresponding angles are congruent, then the two lines are parallel), we can say \(m\parallel n\).

Step1: Recall angle - congruence relationships

Vertical angles are congruent. \(\angle1\) and \(\angle3\) are vertical angles, so \(\angle1\cong\angle3\).
Corresponding angles are congruent when two lines are parallel. \(\angle1\) and \(\angle5\) are corresponding angles, so \(\angle1\cong\angle5\).
Also, \(\angle1\) and \(\angle7\) are alternate - exterior angles, so \(\angle1\cong\angle7\)

Answer:

\(b\parallel c\)

5.