Question

for #1 - 5, given that m || n, state the name of the angle pair. then circle/highlight whether they are congruent or supplementary. write an equation on how you would set - up to solve. the first one is completed for you as an example.

1. \\( \angle 4 \\) and \\( \angle 8 \\) corresponding angles

equation: \\( \angle 4 \cong \angle 8 \\) or supplementary

1. \\( \angle 6 \\) and \\( \angle 7 \\)

equation: \\( \cong \\) or supplementary

1. \\( \angle 5 \\) and \\( \angle 7 \\)

equation: \\( \cong \\) or supplementary

1. \\( \angle 5 \\) and \\( \angle 1 \\)

equation: \\( \cong \\) or supplementary

1. \\( \angle 1 \\) and \\( \angle 4 \\)

equation: \\( \cong \\) or supplementary
find the measures of all the missing angles. (there are 7 in each diagram)
6)
7)
8)

Explanation:

Step1: Identify angle relationships

When two parallel lines \(m\parallel n\) are cut by a transversal, corresponding angles are congruent (\(\angle4\cong\angle8\)), alternate - interior angles are congruent, and consecutive - interior angles are supplementary.

Step2: Analyze each pair

• For \(\angle4\) and \(\angle8\):

Since \(m\parallel n\) and they are corresponding angles, the equation is \(\angle4\cong\angle8\).

• For \(\angle6\) and \(\angle7\):

They are consecutive - interior angles. So the equation is \(\angle6+\angle7 = 180^{\circ}\) (supplementary).

• For \(\angle5\) and \(\angle7\):

They are alternate - interior angles. So the equation is \(\angle5\cong\angle7\).

• For \(\angle5\) and \(\angle1\):

They are corresponding angles. So the equation is \(\angle5\cong\angle1\).

• For \(\angle1\) and \(\angle4\):

They are consecutive - interior angles. So the equation is \(\angle1+\angle4=180^{\circ}\) (supplementary).

Step3: Solve for angles in diagrams 6 - 8

• Diagram 6:

The given angle is \(132^{\circ}\). Let's assume the given angle and one of the angles formed by the transversal with the parallel lines. If the given angle and an angle \(x\) are supplementary (consecutive - interior or linear - pair), \(x = 180 - 132=48^{\circ}\). Using angle - congruence (corresponding, alternate - interior) and supplementary relationships, all angles: \(132^{\circ},48^{\circ},132^{\circ},48^{\circ},132^{\circ},48^{\circ},132^{\circ}\) (depending on the position of the transversal and parallel lines).

• Diagram 7:

The given angle is \(72^{\circ}\). If the given angle and an angle \(y\) are supplementary (consecutive - interior or linear - pair), \(y = 180 - 72 = 108^{\circ}\). Using angle - congruence (corresponding, alternate - interior) and supplementary relationships, all angles: \(72^{\circ},108^{\circ},72^{\circ},108^{\circ},72^{\circ},108^{\circ},72^{\circ}\).

• Diagram 8:

The given angle is \(67^{\circ}\). If the given angle and an angle \(z\) are supplementary (consecutive - interior or linear - pair), \(z=180 - 67 = 113^{\circ}\). Using angle - congruence (corresponding, alternate - interior) and supplementary relationships, all angles: \(67^{\circ},113^{\circ},67^{\circ},113^{\circ},67^{\circ},113^{\circ},67^{\circ}\).

Answer:

1. Equation: \(\angle4\cong\angle8\)
2. Equation: \(\angle6+\angle7 = 180^{\circ}\)
3. Equation: \(\angle5\cong\angle7\)
4. Equation: \(\angle5\cong\angle1\)
5. Equation: \(\angle1+\angle4 = 180^{\circ}\)
6. Angle measures: \(132^{\circ},48^{\circ},132^{\circ},48^{\circ},132^{\circ},48^{\circ},132^{\circ}\)
7. Angle measures: \(72^{\circ},108^{\circ},72^{\circ},108^{\circ},72^{\circ},108^{\circ},72^{\circ}\)
8. Angle measures: \(67^{\circ},113^{\circ},67^{\circ},113^{\circ},67^{\circ},113^{\circ},67^{\circ}\)