Question

e. in the diagram, transversal ( t ) intersects parallel lines ( m ) and ( n ). suppose that the measure of ( angle 4 ) is ( 106^{circ} ). classify the given angle pair. then determine each measure.

1. ( angle 4 ) and ( angle 1, mangle 1=)
2. ( angle 4 ) and ( angle 2, mangle 2=)
3. ( angle 4 ) and ( angle 3, mangle 3=)
4. ( angle 4 ) and ( angle 8, mangle 8=)
5. ( angle 4 ) and ( angle 5, mangle 5=)
6. ( angle 5 ) and ( angle 7, mangle 7=)

Explanation:

Step1: Classify ∠4 and ∠1

∠4 and ∠1 are supplementary angles (they form a linear pair).
$$m\angle1 = 180^{\circ}-m\angle4$$
$$m\angle1 = 180^{\circ}- 106^{\circ}=74^{\circ}$$

Step2: Classify ∠4 and ∠2

∠4 and ∠2 are vertical angles. Vertical angles are equal.
$$m\angle2=m\angle4 = 106^{\circ}$$

Step3: Classify ∠4 and ∠3

∠4 and ∠3 are supplementary angles (they form a linear pair).
$$m\angle3 = 180^{\circ}-m\angle4$$
$$m\angle3=180^{\circ}-106^{\circ} = 74^{\circ}$$

Step4: Classify ∠4 and ∠8

Since lines \(m\parallel n\) and \(t\) is a transversal, ∠4 and ∠8 are corresponding angles. Corresponding angles are equal when lines are parallel.
$$m\angle8=m\angle4 = 106^{\circ}$$

Step5: Classify ∠4 and ∠5

Since lines \(m\parallel n\) and \(t\) is a transversal, ∠4 and ∠5 are alternate - interior angles. Alternate - interior angles are equal when lines are parallel.
$$m\angle5=m\angle4 = 106^{\circ}$$

Step6: Classify ∠5 and ∠7

∠5 and ∠7 are vertical angles. Vertical angles are equal.
$$m\angle7=m\angle5 = 106^{\circ}$$

Answer:

1. \(m\angle1 = 74^{\circ}\)
2. \(m\angle2 = 106^{\circ}\)
3. \(m\angle3 = 74^{\circ}\)
4. \(m\angle8 = 106^{\circ}\)
5. \(m\angle5 = 106^{\circ}\)
6. \(m\angle7 = 106^{\circ}\)