Parallel, Intersecting, and Perpendicular Lines Worksheets - Free Printable

Problem Description:

The task involves analyzing a geometric figure with intersecting lines and angles. The goal is to determine the measures of various angles based on the given information and properties of parallel lines and transversals.

Given Information:

1. Line $ l_1 $ is parallel to line $ l_2 $.
2. A transversal intersects these parallel lines, creating several angles.
3. Some angle measures are provided:
- Angle 1: $ 60^\circ $
- Angle 2: $ 90^\circ $
- Angle 3: $ 30^\circ $

Task:

Determine the measures of all the labeled angles (Angle 1 through Angle 10).

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Solution:

#### Step 1: Understand the Properties of Parallel Lines and Transversals
When a transversal intersects two parallel lines:
- Corresponding Angles are equal.
- Alternate Interior Angles are equal.
- Alternate Exterior Angles are equal.
- Consecutive Interior Angles (same-side interior angles) are supplementary (sum to $ 180^\circ $).
- Vertical Angles are equal.

#### Step 2: Analyze the Given Angles
From the diagram:
- Angle 1: $ 60^\circ $
- Angle 2: $ 90^\circ $
- Angle 3: $ 30^\circ $

#### Step 3: Determine Other Angles Using Geometric Properties

##### Angle 4:
- Angle 4 is a vertical angle to Angle 1.
- Vertical angles are equal.
- Therefore, Angle 4 = $ 60^\circ $.

##### Angle 5:
- Angle 5 is an alternate interior angle to Angle 3.
- Alternate interior angles are equal when the lines are parallel.
- Therefore, Angle 5 = $ 30^\circ $.

##### Angle 6:
- Angle 6 is a consecutive interior angle to Angle 3.
- Consecutive interior angles are supplementary.
- Therefore, Angle 6 = $ 180^\circ - 30^\circ = 150^\circ $.

##### Angle 7:
- Angle 7 is a vertical angle to Angle 6.
- Vertical angles are equal.
- Therefore, Angle 7 = $ 150^\circ $.

##### Angle 8:
- Angle 8 is an alternate interior angle to Angle 2.
- Alternate interior angles are equal when the lines are parallel.
- Therefore, Angle 8 = $ 90^\circ $.

##### Angle 9:
- Angle 9 is a consecutive interior angle to Angle 2.
- Consecutive interior angles are supplementary.
- Therefore, Angle 9 = $ 180^\circ - 90^\circ = 90^\circ $.

##### Angle 10:
- Angle 10 is a vertical angle to Angle 9.
- Vertical angles are equal.
- Therefore, Angle 10 = $ 90^\circ $.

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Final Answer:

The measures of the angles are:
1. Angle 1: $ 60^\circ $
2. Angle 2: $ 90^\circ $
3. Angle 3: $ 30^\circ $
4. Angle 4: $ 60^\circ $
5. Angle 5: $ 30^\circ $
6. Angle 6: $ 150^\circ $
7. Angle 7: $ 150^\circ $
8. Angle 8: $ 90^\circ $
9. Angle 9: $ 90^\circ $
10. Angle 10: $ 90^\circ $

$$
\boxed{60^\circ, 90^\circ, 30^\circ, 60^\circ, 30^\circ, 150^\circ, 150^\circ, 90^\circ, 90^\circ, 90^\circ}
$$