Question
Examine the diagram. It is not drawn to scale.
A triangle has angles A, B, 80 degrees. The exterior angle to angle A is 150 degrees.
Choose the statements that are true based on the diagram.
∠A is supplementary to the exterior angle.
∠A and the 80° angle are remote interior angles.
80° + m∠B + 150° = 360°.
∠B measures 70°.
∠A measures 40°.
A triangle has angles A, B, 80 degrees. The exterior angle to angle A is 150 degrees.
Choose the statements that are true based on the diagram.
∠A is supplementary to the exterior angle.
∠A and the 80° angle are remote interior angles.
80° + m∠B + 150° = 360°.
∠B measures 70°.
∠A measures 40°.
Answers
GPT-4o mini
Based on the provided information about the triangle and the exterior angle, we can analyze each statement one by one:
1. **∠A is supplementary to the exterior angle.**
- True. An exterior angle is supplementary to the adjacent interior angle of the triangle. Therefore, ∠A + 150° = 180°, which means ∠A is supplementary to the exterior angle.
2. **∠A and the 80° angle are remote interior angles.**
- True. In a triangle, the two interior angles that are not adjacent to a given exterior angle are considered remote interior angles. Since the exterior angle given (150°) is adjacent to ∠B, both ∠A and the 80° angle are remote from this exterior angle.
3. **80° + m∠B + 150° = 360°.**
- False. The sum of angles around a point is 360°, but this equation includes the exterior angle and one interior angle. The equation should reflect that the sum of the angles of a triangle is 180°. Specifically, it should be 80° + m∠A + m∠B = 180°.
4. **∠B measures 70°.**
- True. Since the exterior angle is equal to the sum of the two remote interior angles (∠A and the angle of 80°), we have:
\[
150° = ∠A + 80°.
\]
From the complementary interior angle relationship, we can find m∠B:
\[
∠A + ∠B + 80° = 180°.
\]
Since ∠A = 30°, that makes ∠B = 70°.
5. **∠A measures 40°.**
- False. Based on the calculation from the exterior angle, ∠A = 30° because:
\[
∠A + 80° = 150° \Rightarrow ∠A = 150° - 80° = 70°.
\]
### Conclusion:
The true statements based on the information provided are:
- ∠A is supplementary to the exterior angle.
- ∠A and the 80° angle are remote interior angles.
- ∠B measures 70°.
1. **∠A is supplementary to the exterior angle.**
- True. An exterior angle is supplementary to the adjacent interior angle of the triangle. Therefore, ∠A + 150° = 180°, which means ∠A is supplementary to the exterior angle.
2. **∠A and the 80° angle are remote interior angles.**
- True. In a triangle, the two interior angles that are not adjacent to a given exterior angle are considered remote interior angles. Since the exterior angle given (150°) is adjacent to ∠B, both ∠A and the 80° angle are remote from this exterior angle.
3. **80° + m∠B + 150° = 360°.**
- False. The sum of angles around a point is 360°, but this equation includes the exterior angle and one interior angle. The equation should reflect that the sum of the angles of a triangle is 180°. Specifically, it should be 80° + m∠A + m∠B = 180°.
4. **∠B measures 70°.**
- True. Since the exterior angle is equal to the sum of the two remote interior angles (∠A and the angle of 80°), we have:
\[
150° = ∠A + 80°.
\]
From the complementary interior angle relationship, we can find m∠B:
\[
∠A + ∠B + 80° = 180°.
\]
Since ∠A = 30°, that makes ∠B = 70°.
5. **∠A measures 40°.**
- False. Based on the calculation from the exterior angle, ∠A = 30° because:
\[
∠A + 80° = 150° \Rightarrow ∠A = 150° - 80° = 70°.
\]
### Conclusion:
The true statements based on the information provided are:
- ∠A is supplementary to the exterior angle.
- ∠A and the 80° angle are remote interior angles.
- ∠B measures 70°.