Question
Examine the diagram. It is not drawn to scale.
A triangle has angles 1, 2, 3. A line extends from 2 through 1 to form exterior angle 5. A line extends from 2 through 3 to form exterior angle 4.
Which statements are true based on the given diagram? Select all that apply.
∠1 is a remote interior angle to exterior ∠5.
∠2 is a remote interior angle to both exterior angles 4 and 5.
∠3 is an adjacent interior angle to exterior ∠4.
∠1 and ∠5 are supplementary angles.
∠2 and ∠4 are congruent angles.
A triangle has angles 1, 2, 3. A line extends from 2 through 1 to form exterior angle 5. A line extends from 2 through 3 to form exterior angle 4.
Which statements are true based on the given diagram? Select all that apply.
∠1 is a remote interior angle to exterior ∠5.
∠2 is a remote interior angle to both exterior angles 4 and 5.
∠3 is an adjacent interior angle to exterior ∠4.
∠1 and ∠5 are supplementary angles.
∠2 and ∠4 are congruent angles.
Answers
GPT-4o mini
To analyze the relationships between the angles in the triangle and the exterior angles, we can use the properties of angles in triangles and exterior angle theory.
1. **∠1 is a remote interior angle to exterior ∠5.**
- *True.* Remote interior angles are the angles inside the triangle that are not adjacent to the exterior angle. Since ∠5 is formed by extending the line from vertex 2 through 1, it is an exterior angle corresponding to angle 3. Therefore, ∠1 is a remote interior angle relative to ∠5.
2. **∠2 is a remote interior angle to both exterior angles 4 and 5.**
- *False.* ∠2 is the angle at vertex 2. It is adjacent to exterior angle ∠4 (formed by extending the line through ∠2 and angle 3), so it cannot be considered a remote interior angle to ∠4. Therefore, this statement is false.
3. **∠3 is an adjacent interior angle to exterior ∠4.**
- *True.* An adjacent angle is one that shares a vertex and one side with another angle. Since ∠4 is formed by extending the line from angle 2 through angle 3, ∠3 is indeed adjacent to ∠4.
4. **∠1 and ∠5 are supplementary angles.**
- *True.* Exterior angle ∠5 is equal to the sum of the two remote interior angles, which are ∠1 and ∠3. Therefore, as ∠1 and ∠5 sum to 180 degrees, they are supplementary.
5. **∠2 and ∠4 are congruent angles.**
- *False.* Since ∠4 is an exterior angle formed by extending the triangle at vertex 2 through angle 3, it generally would not equal the interior angle ∠2 unless specific conditions (like an isosceles triangle) hold. Without any information suggesting that this triangle is isosceles, we cannot conclude they are congruent.
Based on this analysis, the true statements are:
- **∠1 is a remote interior angle to exterior ∠5.**
- **∠3 is an adjacent interior angle to exterior ∠4.**
- **∠1 and ∠5 are supplementary angles.**
1. **∠1 is a remote interior angle to exterior ∠5.**
- *True.* Remote interior angles are the angles inside the triangle that are not adjacent to the exterior angle. Since ∠5 is formed by extending the line from vertex 2 through 1, it is an exterior angle corresponding to angle 3. Therefore, ∠1 is a remote interior angle relative to ∠5.
2. **∠2 is a remote interior angle to both exterior angles 4 and 5.**
- *False.* ∠2 is the angle at vertex 2. It is adjacent to exterior angle ∠4 (formed by extending the line through ∠2 and angle 3), so it cannot be considered a remote interior angle to ∠4. Therefore, this statement is false.
3. **∠3 is an adjacent interior angle to exterior ∠4.**
- *True.* An adjacent angle is one that shares a vertex and one side with another angle. Since ∠4 is formed by extending the line from angle 2 through angle 3, ∠3 is indeed adjacent to ∠4.
4. **∠1 and ∠5 are supplementary angles.**
- *True.* Exterior angle ∠5 is equal to the sum of the two remote interior angles, which are ∠1 and ∠3. Therefore, as ∠1 and ∠5 sum to 180 degrees, they are supplementary.
5. **∠2 and ∠4 are congruent angles.**
- *False.* Since ∠4 is an exterior angle formed by extending the triangle at vertex 2 through angle 3, it generally would not equal the interior angle ∠2 unless specific conditions (like an isosceles triangle) hold. Without any information suggesting that this triangle is isosceles, we cannot conclude they are congruent.
Based on this analysis, the true statements are:
- **∠1 is a remote interior angle to exterior ∠5.**
- **∠3 is an adjacent interior angle to exterior ∠4.**
- **∠1 and ∠5 are supplementary angles.**