Question
Parallel lines r and s are cut by two transversals, parallel lines t and u.
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 r and u, 13, 14, 15, 16.
How many angles are alternate exterior angles with angle 5?
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 r and u, 13, 14, 15, 16.
How many angles are alternate exterior angles with angle 5?
Answers
GPT-4o mini
To determine how many angles are alternate exterior angles with angle \(5\), we need to first understand the arrangement of the angles formed by the parallel lines and transversals.
Given the following angles at the intersections:
- \(r\) and \(t\): angles \(1, 2, 3, 4\) (clockwise)
- \(s\) and \(t\): angles \(5, 6, 7, 8\) (clockwise)
- \(s\) and \(u\): angles \(9, 10, 11, 12\) (clockwise)
- \(r\) and \(u\): angles \(13, 14, 15, 16\) (clockwise)
### Identifying Alternate Exterior Angles
Alternate exterior angles are pairs of angles that are on opposite sides of the transversal and outside the two parallel lines.
1. **Identifying the parallel lines**:
- Lines \(r\) and \(s\) are parallel.
- Lines \(t\) and \(u\) are the transversals.
2. **Position of angle \(5\)**:
- Angle \(5\) is at the intersection of lines \(s\) (the bottom parallel line) and \(t\) (the transversal), positioned at the top right of the intersection of \(s\) and \(t\).
3. **Identifying alternate exterior angles** with respect to angle \(5\):
- Angle \(5\) has exterior angles formed by the lines:
- \(\text{Exterior to line } r\) above \(r\)
- \(\text{Exterior to line } s\) on the same side as angle \(5\)
The angles that are alternate exterior to angle \(5\) with respect to the parallel lines \(r\) and \(s\) are:
- Angle \(12\): This angle is at the intersection of \(s\) and \(u\) (well below), and is on the opposite side of angle \(5\) across line \(s\).
- Angle \(16\): This angle is at the intersection of \(r\) and \(u\) (well below) and is on the opposite side of angle \(5\) across line \(r\).
### Conclusion
The two angles that are alternate exterior angles with angle \(5\) are angles:
- \(12\)
- \(16\)
Despite a small misinterpretation earlier, the correct identification leads us to confirm that:
\[
\text{There are } 2 \text{ angles that are alternate exterior angles with angle } 5.
\]
Thus, the answer is:
\[
\boxed{2}
\]
Given the following angles at the intersections:
- \(r\) and \(t\): angles \(1, 2, 3, 4\) (clockwise)
- \(s\) and \(t\): angles \(5, 6, 7, 8\) (clockwise)
- \(s\) and \(u\): angles \(9, 10, 11, 12\) (clockwise)
- \(r\) and \(u\): angles \(13, 14, 15, 16\) (clockwise)
### Identifying Alternate Exterior Angles
Alternate exterior angles are pairs of angles that are on opposite sides of the transversal and outside the two parallel lines.
1. **Identifying the parallel lines**:
- Lines \(r\) and \(s\) are parallel.
- Lines \(t\) and \(u\) are the transversals.
2. **Position of angle \(5\)**:
- Angle \(5\) is at the intersection of lines \(s\) (the bottom parallel line) and \(t\) (the transversal), positioned at the top right of the intersection of \(s\) and \(t\).
3. **Identifying alternate exterior angles** with respect to angle \(5\):
- Angle \(5\) has exterior angles formed by the lines:
- \(\text{Exterior to line } r\) above \(r\)
- \(\text{Exterior to line } s\) on the same side as angle \(5\)
The angles that are alternate exterior to angle \(5\) with respect to the parallel lines \(r\) and \(s\) are:
- Angle \(12\): This angle is at the intersection of \(s\) and \(u\) (well below), and is on the opposite side of angle \(5\) across line \(s\).
- Angle \(16\): This angle is at the intersection of \(r\) and \(u\) (well below) and is on the opposite side of angle \(5\) across line \(r\).
### Conclusion
The two angles that are alternate exterior angles with angle \(5\) are angles:
- \(12\)
- \(16\)
Despite a small misinterpretation earlier, the correct identification leads us to confirm that:
\[
\text{There are } 2 \text{ angles that are alternate exterior angles with angle } 5.
\]
Thus, the answer is:
\[
\boxed{2}
\]