Question
If Angle 6 is congruent to angle 10 and Angle 5 is congruent to angle 7, which describes all the lines that must be parallel?
Lines r and s are crossed by lines t and u to form 16 angles. Clockwise from top left, at the intersection of r and t, the angles are 1, 2, 3, 4; at the intersection of s and t, 5, 6, 7, 8; at the intersection of u and s, 9, 10, 11, 12; at the intersection of u and r, 13, 14, 15, 16.
Only lines r and s must be parallel.
Only lines t and u must be parallel.
Lines r and s and lines t and u must be parallel.
Neither lines r and s nor lines t and u must be parallel.
Lines r and s are crossed by lines t and u to form 16 angles. Clockwise from top left, at the intersection of r and t, the angles are 1, 2, 3, 4; at the intersection of s and t, 5, 6, 7, 8; at the intersection of u and s, 9, 10, 11, 12; at the intersection of u and r, 13, 14, 15, 16.
Only lines r and s must be parallel.
Only lines t and u must be parallel.
Lines r and s and lines t and u must be parallel.
Neither lines r and s nor lines t and u must be parallel.
Answers
GPT-4o mini
To determine which lines must be parallel based on the information given about angles 5, 6, 7, and 10, we can utilize the properties of angles formed when two transversal lines intersect parallel lines.
Given:
- Angle 6 is congruent to angle 10.
- Angle 5 is congruent to angle 7.
1. **Angle Relationships**:
- Angles 6 and 10 are corresponding angles created by lines t and u intersecting the lines r and s. If these angles are congruent, this implies that lines r and s must be parallel for the corresponding angles postulate to hold.
- Angles 5 and 7 are also corresponding angles created by lines t and u intersecting lines s and r. Since these angles are congruent, this also reinforces that lines r and s must be parallel.
Given that congruent corresponding angles indicate that the lines cut by a transversal are parallel, we can conclude that lines r and s must be parallel.
2. **Lines t and u**:
- The congruence of angles 6 and 10 and of angles 5 and 7 does not provide information about the relationship between lines t and u directly.
After analyzing the relationships, we arrive at the conclusion that the only lines that must be parallel are lines r and s.
Therefore, the answer is:
**Only lines r and s must be parallel.**
Given:
- Angle 6 is congruent to angle 10.
- Angle 5 is congruent to angle 7.
1. **Angle Relationships**:
- Angles 6 and 10 are corresponding angles created by lines t and u intersecting the lines r and s. If these angles are congruent, this implies that lines r and s must be parallel for the corresponding angles postulate to hold.
- Angles 5 and 7 are also corresponding angles created by lines t and u intersecting lines s and r. Since these angles are congruent, this also reinforces that lines r and s must be parallel.
Given that congruent corresponding angles indicate that the lines cut by a transversal are parallel, we can conclude that lines r and s must be parallel.
2. **Lines t and u**:
- The congruence of angles 6 and 10 and of angles 5 and 7 does not provide information about the relationship between lines t and u directly.
After analyzing the relationships, we arrive at the conclusion that the only lines that must be parallel are lines r and s.
Therefore, the answer is:
**Only lines r and s must be parallel.**