Question

1. find m∠1 and m∠2. tell which theorem you would use in each case.

2.

Explanation:

Step1: Identify angle relationships for \(\angle1\) in the first case

Use corresponding - angles theorem. Since the lines are parallel and \(\angle1\) is corresponding to the \(117^{\circ}\) angle, \(m\angle1 = 117^{\circ}\).

Step2: Identify angle relationship for \(\angle2\) in the first case

Use vertical - angles property. Since \(\angle1\) and \(\angle2\) are vertical angles, \(m\angle2=m\angle1 = 117^{\circ}\).

Step3: Identify angle relationships for \(\angle1\) in the second case

Use alternate - exterior angles theorem. Since the lines are parallel and \(\angle1\) is alternate - exterior to the \(132^{\circ}\) angle, \(m\angle1 = 132^{\circ}\).

Step4: Identify angle relationship for \(\angle2\) in the second case

Use vertical - angles property. Since \(\angle1\) and \(\angle2\) are vertical angles, \(m\angle2=m\angle1 = 132^{\circ}\).

Answer:

1.

• \(m\angle1 = 117^{\circ}\), \(m\angle2 = 117^{\circ}\), Theorem: Corresponding - angles theorem.
• Explanation: When two parallel lines are cut by a transversal, corresponding angles are congruent. \(\angle1\) and the given \(117^{\circ}\) angle are corresponding angles, so \(m\angle1 = 117^{\circ}\). Also, \(\angle1\) and \(\angle2\) are vertical angles, and vertical angles are congruent, so \(m\angle2=m\angle1 = 117^{\circ}\).

2.

• \(m\angle1 = 132^{\circ}\), \(m\angle2 = 132^{\circ}\), Theorem: Alternate - exterior angles theorem.
• Explanation: When two parallel lines are cut by a transversal, alternate - exterior angles are congruent. \(\angle1\) and the given \(132^{\circ}\) angle are alternate - exterior angles, so \(m\angle1 = 132^{\circ}\). Also, \(\angle1\) and \(\angle2\) are vertical angles, and vertical angles are congruent, so \(m\angle2=m\angle1 = 132^{\circ}\).