Question
Line p is parallel to line q.
Parallel lines p and q are crossed by lines a and b to form 2 triangles. At parallel line p, angle 4 is formed by line b and angle 5 is formed by line a. Angle 6 is the third angle. At parallel line q, angle 3 is formed by line 3 and angle 2 is formed by line b. Angle 1 is the third angle.
Which set of statements about the angles is true?
Angle 1 is congruent to angle 6, angle 5 is congruent to angle 4, angle 3 is congruent to angle 2
Angle 2 is congruent to angle 4, angle 3 is congruent to angle 6, angle 1 is congruent to angle 5
Angle 3 is congruent to angle 6, angle 1 is congruent to angle 2, angle 5 is congruent to angle 4
Angle 6 is congruent to angle 1, angle 5 is congruent to angle 3, angle 4 is congruent to angle 2
Parallel lines p and q are crossed by lines a and b to form 2 triangles. At parallel line p, angle 4 is formed by line b and angle 5 is formed by line a. Angle 6 is the third angle. At parallel line q, angle 3 is formed by line 3 and angle 2 is formed by line b. Angle 1 is the third angle.
Which set of statements about the angles is true?
Angle 1 is congruent to angle 6, angle 5 is congruent to angle 4, angle 3 is congruent to angle 2
Angle 2 is congruent to angle 4, angle 3 is congruent to angle 6, angle 1 is congruent to angle 5
Angle 3 is congruent to angle 6, angle 1 is congruent to angle 2, angle 5 is congruent to angle 4
Angle 6 is congruent to angle 1, angle 5 is congruent to angle 3, angle 4 is congruent to angle 2
Answers
GPT-4o mini
Given that lines p and q are parallel and are crossed by lines a and b, we can use the properties of angles formed by parallel lines and transversals to determine the relationships between the angles.
1. **Angle Relationships**:
- **Angles on the same side of a transversal**: Angles that are on the same side of the transversal and inside the parallel lines are supplementary.
- **Alternate interior angles**: Angles that are on opposite sides of the transversal and inside the parallel lines are congruent.
- **Corresponding angles**: Angles that are on the same side of the transversal and in corresponding positions are congruent.
Based on these properties, let's define the angles formed:
- **At line p**:
- Angle 4 (between line b and line p)
- Angle 5 (between line a and line p)
- Angle 6 (the third angle formed at line p)
- **At line q**:
- Angle 1 (the third angle formed at line q)
- Angle 2 (between line b and line q)
- Angle 3 (between line a and line q)
2. **Finding Relations**:
- **Angle 1 is congruent to Angle 6**: Since they are alternate interior angles (formed by lines a with parallel lines p and q).
- **Angle 5 is congruent to Angle 4**: These are corresponding angles formed by line b on parallel lines p and q.
- **Angle 2 is congruent to Angle 4** and **Angle 3 is congruent to Angle 6**: Angle 2 is supplementary to angle 5, and angle 3 is supplementary to angle 6. So these pairs do not represent parallel angles.
Therefore, based on the angle relationships derived from the properties of parallel lines and transversals, the correct set of statements from your options is:
**Angle 1 is congruent to angle 6, angle 5 is congruent to angle 4, angle 3 is congruent to angle 2**.
Thus, this leads to the conclusion that the first option is correct:
**Angle 1 is congruent to angle 6, angle 5 is congruent to angle 4, angle 3 is congruent to angle 2.**
1. **Angle Relationships**:
- **Angles on the same side of a transversal**: Angles that are on the same side of the transversal and inside the parallel lines are supplementary.
- **Alternate interior angles**: Angles that are on opposite sides of the transversal and inside the parallel lines are congruent.
- **Corresponding angles**: Angles that are on the same side of the transversal and in corresponding positions are congruent.
Based on these properties, let's define the angles formed:
- **At line p**:
- Angle 4 (between line b and line p)
- Angle 5 (between line a and line p)
- Angle 6 (the third angle formed at line p)
- **At line q**:
- Angle 1 (the third angle formed at line q)
- Angle 2 (between line b and line q)
- Angle 3 (between line a and line q)
2. **Finding Relations**:
- **Angle 1 is congruent to Angle 6**: Since they are alternate interior angles (formed by lines a with parallel lines p and q).
- **Angle 5 is congruent to Angle 4**: These are corresponding angles formed by line b on parallel lines p and q.
- **Angle 2 is congruent to Angle 4** and **Angle 3 is congruent to Angle 6**: Angle 2 is supplementary to angle 5, and angle 3 is supplementary to angle 6. So these pairs do not represent parallel angles.
Therefore, based on the angle relationships derived from the properties of parallel lines and transversals, the correct set of statements from your options is:
**Angle 1 is congruent to angle 6, angle 5 is congruent to angle 4, angle 3 is congruent to angle 2**.
Thus, this leads to the conclusion that the first option is correct:
**Angle 1 is congruent to angle 6, angle 5 is congruent to angle 4, angle 3 is congruent to angle 2.**