Question

tell which angles are congruent to the given angle measure.
1
m||n
<2, <5, <6
tell why each of the angles indicated are congruent to the angle measure shown.
3
m||n
∠3:
∠4:
∠7:
find the indicated angle measures. then justify each answer.
5
m∠1 =
6
m∠1 =

Explanation:

Step1: Recall angle - relationships in parallel lines

When two parallel lines are cut by a transversal, vertical angles are congruent, corresponding angles are congruent, alternate - interior angles are congruent, and alternate - exterior angles are congruent.

Step2: Analyze problem 1

Given \(m\parallel n\) and an angle of \(100^{\circ}\). \(\angle2\) is a vertical angle to the given angle, so \(\angle2\) is congruent. \(\angle5\) is an alternate - interior angle to the given angle, so \(\angle5\) is congruent. \(\angle6\) is a corresponding angle to the given angle, so \(\angle6\) is congruent.

Step3: Analyze problem 2

For the non - parallel line case, we use vertical angle relationships. If one angle is given, its vertical angle is congruent. So if an angle is given, the angles that are vertical to it are congruent.

Step4: Analyze problem 3

Given \(m\parallel n\) and an angle of \(45^{\circ}\). \(\angle3\) is a vertical angle to the given angle, so \(\angle3\) is congruent. \(\angle4\) is an alternate - interior angle to \(\angle3\), so \(\angle4\) is congruent to the given angle. \(\angle7\) is a vertical angle to \(\angle4\), so \(\angle7\) is congruent to the given angle.

Step5: Analyze problem 4

Given \(m\parallel n\) and an angle of \(130^{\circ}\). \(\angle1\) is a vertical angle to the given angle, so \(\angle1\) is congruent. \(\angle4\) is an alternate - interior angle to \(\angle1\), so \(\angle4\) is congruent. \(\angle7\) is a vertical angle to \(\angle4\), so \(\angle7\) is congruent.

Step6: Analyze problem 5

\(\angle1\) and the \(115^{\circ}\) angle are vertical angles. Since vertical angles are congruent, \(m\angle1 = 115^{\circ}\).

Step7: Analyze problem 6

\(\angle1\) and the \(84^{\circ}\) angle are vertical angles. Since vertical angles are congruent, \(m\angle1=84^{\circ}\).

Answer:

1. Congruent angles to the given \(100^{\circ}\) angle are \(\angle2\), \(\angle5\), \(\angle6\) because of vertical - angle, alternate - interior, and corresponding - angle relationships respectively.
2. Use vertical - angle relationships to find congruent angles.
3. \(\angle3\) is congruent because it's a vertical angle, \(\angle4\) is congruent because it's an alternate - interior angle to \(\angle3\), \(\angle7\) is congruent because it's a vertical angle to \(\angle4\).
4. \(\angle1\) is congruent because it's a vertical angle, \(\angle4\) is congruent because it's an alternate - interior angle to \(\angle1\), \(\angle7\) is congruent because it's a vertical angle to \(\angle4\).
5. \(m\angle1 = 115^{\circ}\) because \(\angle1\) and the \(115^{\circ}\) angle are vertical angles.
6. \(m\angle1 = 84^{\circ}\) because \(\angle1\) and the \(84^{\circ}\) angle are vertical angles.