Question

given the following diagram, find the measure of each angle. justify your answer with their relationship.

1. if the measure of ∠7 = 45°, then the measure of ∠5 =

what is the relationship between the angles?

1. if the measure of ∠2 = 50°, then the measure of ∠6 =

what is the relationship between the angles?

1. if the measure of ∠1 = 140°, then the measure of ∠5 =

what is the relationship between the angles?

1. if the measure of ∠7 = 20°, then the measure of ∠4 =

what is the relationship between the angles?

1. if the measure of ∠6 = 55°, then the measure of ∠7 =

what is the relationship between the angles?

1. if the measure of ∠3 = 160°, then the measure of ∠1 =

what is the relationship between the angles?
if the measure of ∠4 = 125°, then the measure of ∠3 =
what is the relationship between the angles?

Explanation:

1)

Step1: Use linear - pair relationship

Since \(\angle7\) and \(\angle5\) form a linear pair, \(\angle7+\angle5 = 180^{\circ}\). Given \(\angle7 = 45^{\circ}\), then \(\angle5=180^{\circ}-\angle7\).

Step2: Calculate \(\angle5\)

\(\angle5 = 180^{\circ}-45^{\circ}=135^{\circ}\). Relationship: Linear - pair (supplementary angles).

2)

Step1: Use alternate - interior angles relationship (since \(k\parallel d\))

\(\angle2\) and \(\angle6\) are alternate - interior angles. When two parallel lines \(k\) and \(d\) are cut by a transversal, alternate - interior angles are equal. Given \(\angle2 = 50^{\circ}\), so \(\angle6=\angle2\).

Step2: State the measure of \(\angle6\)

\(\angle6 = 50^{\circ}\). Relationship: Alternate - interior angles.

3)

Step1: Use vertical - angles relationship

\(\angle1\) and \(\angle5\) are vertical angles. Vertical angles are equal. Given \(\angle1 = 140^{\circ}\), so \(\angle5=\angle1\).

Step2: State the measure of \(\angle5\)

\(\angle5 = 140^{\circ}\). Relationship: Vertical angles.

4)

Step1: Use alternate - exterior angles relationship (since \(k\parallel d\))

\(\angle7\) and \(\angle4\) are alternate - exterior angles. When two parallel lines \(k\) and \(d\) are cut by a transversal, alternate - exterior angles are equal. Given \(\angle7 = 20^{\circ}\), so \(\angle4=\angle7\).

Step2: State the measure of \(\angle4\)

\(\angle4 = 20^{\circ}\). Relationship: Alternate - exterior angles.

5)

Step1: Use corresponding angles relationship (since \(k\parallel d\))

\(\angle6\) and \(\angle7\) are corresponding angles. When two parallel lines \(k\) and \(d\) are cut by a transversal, corresponding angles are equal. Given \(\angle6 = 55^{\circ}\), so \(\angle7=\angle6\).

Step2: State the measure of \(\angle7\)

\(\angle7 = 55^{\circ}\). Relationship: Corresponding angles.

6)

Step1: Use vertical - angles and corresponding angles relationship

First, \(\angle3\) and \(\angle6\) are vertical angles, so \(\angle6=\angle3 = 160^{\circ}\). Then, since \(k\parallel d\), \(\angle1\) and \(\angle6\) are corresponding angles. So \(\angle1=\angle6\).

Step2: State the measure of \(\angle1\)

\(\angle1 = 160^{\circ}\). Relationship: Vertical angles and corresponding angles.

7)

Step1: Use linear - pair relationship

\(\angle4\) and \(\angle3\) form a linear pair. So \(\angle4+\angle3 = 180^{\circ}\). Given \(\angle4 = 125^{\circ}\), then \(\angle3=180^{\circ}-\angle4\).

Step2: Calculate \(\angle3\)

\(\angle3 = 180^{\circ}-125^{\circ}=55^{\circ}\). Relationship: Linear - pair (supplementary angles).

Answer:

1. \(\angle5 = 135^{\circ}\), Linear - pair.
2. \(\angle6 = 50^{\circ}\), Alternate - interior angles.
3. \(\angle5 = 140^{\circ}\), Vertical angles.
4. \(\angle4 = 20^{\circ}\), Alternate - exterior angles.
5. \(\angle7 = 55^{\circ}\), Corresponding angles.
6. \(\angle1 = 160^{\circ}\), Vertical angles and corresponding angles.
7. \(\angle3 = 55^{\circ}\), Linear - pair.