Let's address the two questions shown in your image.

14. Reasoning Question:

Question: Two horizontal lines are cut by a transversal. What is the least number of angle measures you need to know in order to find the measure of every angle? Explain your reasoning.

Answer: You need to know just one angle in order to find the measure of every angle. This is because of the following angle relationships:

1. Corresponding Angles: When two lines are cut by a transversal, corresponding angles are equal.
2. Alternate Interior and Exterior Angles: These angles are also congruent.
3. Supplementary Angles: Adjacent angles along the transversal add up to 180°.

Knowing one angle (such as any angle formed by the transversal) allows you to use these relationships to deduce the rest of the angles in the system.

15, 16, 17. Finding Measures of Numbered Angles:

Let's examine the specific angles:

Problem 15:

Given angle 5 = 61° (from the diagram):

• Angle 3 (vertical to angle 5) = 61° (Vertical angles are equal).
• Angle 1 (corresponding to angle 5) = 61° (Corresponding angles are congruent).
• Angle 2 (adjacent to angle 1, forming a linear pair) = 180° - 61° = 119° (Linear pairs sum to 180°).
• Angle 4 (adjacent to angle 3, forming a linear pair) = 180° - 61° = 119° (Same linear pair relationship).

Problem 16:

Given angle 5 = 99° (from the diagram):

• Angle 3 (vertical to angle 5) = 99° (Vertical angles are equal).
• Angle 1 (corresponding to angle 5) = 99° (Corresponding angles are congruent).
• Angle 2 (adjacent to angle 1) = 180° - 99° = 81° (Linear pairs).
• Angle 4 (adjacent to angle 3) = 180° - 99° = 81° (Linear pairs).

Problem 17:

We have two perpendicular lines, meaning several angles are right angles (90°):

• Angles 1, 2, 3, and 4 are each 90°, due to the perpendicular nature of the lines.
• Angles 5, 6, 7, and 8 are also 90° for the same reason.

Would you like further clarification on any of the steps or concepts here?

Related Questions:

1. What are alternate interior angles, and how do they relate to transversal lines?
2. How do vertical angles help in determining unknown angle measures?
3. What is the relationship between linear pairs and supplementary angles?
4. How can you determine corresponding angles when a transversal cuts two parallel lines?
5. What are the angle relationships when perpendicular lines intersect?

Tip:

In problems involving a transversal and parallel lines, focus on identifying angle pairs such as corresponding, alternate interior, and alternate exterior angles. This makes solving for unknown angles much quicker.