Question
Line l is parallel to line e in the figure below.
Parallel lines e and l are crossed by lines m and n to form 2 triangles. At the intersection of parallel line e with line n is angle q, and with line m is angle 3. Angle 2 is the third angle. At the intersection of parallel line l and m is 6, at line n is 4. The third angle is 5.
Which statements about the figure are true? Check all that apply.
Vertical angles prove that Angle 1 is congruent to angle 4.
Vertical angles prove that Angle 2 is congruent to angle 5.
The triangles are similar because corresponding sides are congruent.
The triangles are similar because alternate interior angles are congruent.
In the similar triangles, Angle 3 and Angle 6 are alternate interior angles.
In the similar triangles, Angle 3 and Angle 4 are corresponding angles.
Parallel lines e and l are crossed by lines m and n to form 2 triangles. At the intersection of parallel line e with line n is angle q, and with line m is angle 3. Angle 2 is the third angle. At the intersection of parallel line l and m is 6, at line n is 4. The third angle is 5.
Which statements about the figure are true? Check all that apply.
Vertical angles prove that Angle 1 is congruent to angle 4.
Vertical angles prove that Angle 2 is congruent to angle 5.
The triangles are similar because corresponding sides are congruent.
The triangles are similar because alternate interior angles are congruent.
In the similar triangles, Angle 3 and Angle 6 are alternate interior angles.
In the similar triangles, Angle 3 and Angle 4 are corresponding angles.
Answers
GPT-4o mini
Let's analyze the situation involving the triangles formed by the intersecting lines and the relationships between the angles.
1. **Vertical angles prove that Angle 1 is congruent to angle 4.**
- True. Since angles formed by intersecting lines are vertical angles, Angle 1 and Angle 4 are vertical angles and therefore congruent.
2. **Vertical angles prove that Angle 2 is congruent to angle 5.**
- True. Similarly, Angle 2 and Angle 5 are vertical angles and are therefore congruent.
3. **The triangles are similar because corresponding sides are congruent.**
- False. Similarity of triangles is determined by angle relationships, not solely by corresponding sides being congruent.
4. **The triangles are similar because alternate interior angles are congruent.**
- True. Since lines e and l are parallel, and lines m and n intersect them, alternate interior angles formed (like Angle 2 and Angle 5) are congruent, which shows that the triangles are similar.
5. **In the similar triangles, Angle 3 and Angle 6 are alternate interior angles.**
- True. Angle 3 on line e and Angle 6 on line l are alternate interior angles formed by the transversal line m, confirming their congruence and supporting the similarity of the triangles.
6. **In the similar triangles, Angle 3 and Angle 4 are corresponding angles.**
- True. Angle 3 and Angle 4 are corresponding angles since they occupy corresponding positions where the transversal (line m) intersects the parallel lines e and l.
Based on the reasoning provided, the true statements are:
- Vertical angles prove that Angle 1 is congruent to angle 4.
- Vertical angles prove that Angle 2 is congruent to angle 5.
- The triangles are similar because alternate interior angles are congruent.
- In the similar triangles, Angle 3 and Angle 6 are alternate interior angles.
- In the similar triangles, Angle 3 and Angle 4 are corresponding angles.
So the following are true:
1. True
2. True
4. True
5. True
6. True
The third statement about corresponding sides is false.
1. **Vertical angles prove that Angle 1 is congruent to angle 4.**
- True. Since angles formed by intersecting lines are vertical angles, Angle 1 and Angle 4 are vertical angles and therefore congruent.
2. **Vertical angles prove that Angle 2 is congruent to angle 5.**
- True. Similarly, Angle 2 and Angle 5 are vertical angles and are therefore congruent.
3. **The triangles are similar because corresponding sides are congruent.**
- False. Similarity of triangles is determined by angle relationships, not solely by corresponding sides being congruent.
4. **The triangles are similar because alternate interior angles are congruent.**
- True. Since lines e and l are parallel, and lines m and n intersect them, alternate interior angles formed (like Angle 2 and Angle 5) are congruent, which shows that the triangles are similar.
5. **In the similar triangles, Angle 3 and Angle 6 are alternate interior angles.**
- True. Angle 3 on line e and Angle 6 on line l are alternate interior angles formed by the transversal line m, confirming their congruence and supporting the similarity of the triangles.
6. **In the similar triangles, Angle 3 and Angle 4 are corresponding angles.**
- True. Angle 3 and Angle 4 are corresponding angles since they occupy corresponding positions where the transversal (line m) intersects the parallel lines e and l.
Based on the reasoning provided, the true statements are:
- Vertical angles prove that Angle 1 is congruent to angle 4.
- Vertical angles prove that Angle 2 is congruent to angle 5.
- The triangles are similar because alternate interior angles are congruent.
- In the similar triangles, Angle 3 and Angle 6 are alternate interior angles.
- In the similar triangles, Angle 3 and Angle 4 are corresponding angles.
So the following are true:
1. True
2. True
4. True
5. True
6. True
The third statement about corresponding sides is false.