Question
Example
Payne notices that his mom's hourglass resembles two triangles. He sketches the hourglass. Payne's mom considers his sketch. She tells him that she knows KLENM, ZLZM and ZKRL=/NRM. She asks Payne to justify why R must be the midpoint of both KN and ML. How could Payne do this?
L
= MULTIPLE-CHOICE QUESTION
Are the two triangles congruent? How
do you know?
vertical wangles
N
Yes, AAS
Yes, ASA
M
Yes, SSS
Yes, SAS
No
Payne notices that his mom's hourglass resembles two triangles. He sketches the hourglass. Payne's mom considers his sketch. She tells him that she knows KLENM, ZLZM and ZKRL=/NRM. She asks Payne to justify why R must be the midpoint of both KN and ML. How could Payne do this?
L
= MULTIPLE-CHOICE QUESTION
Are the two triangles congruent? How
do you know?
vertical wangles
N
Yes, AAS
Yes, ASA
M
Yes, SSS
Yes, SAS
No
Answers
GPT-4o mini
To determine whether the two triangles (let's assume they are triangle KLM and triangle NRM, based on the provided letters) are congruent and whether R must be the midpoint of both KN and ML, we need to analyze the information given about the angles and sides.
Here's the reasoning:
1. **Identifying Angles and Sides**: According to the problem, we understand Payne's mom has provided some angle relationships and possibly side relationships, represented by KLENM, ZLZM, and ZKRL=/NRM. Although these relationships are not fully spelled out, it suggests that the angles are either equal or have a specific relationship.
2. **Criteria for Triangle Congruence**:
- **ASA (Angle-Side-Angle)**: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
- **AAS (Angle-Angle-Side)**: If two angles and a non-included side of one triangle are equal to two angles and a corresponding side of another triangle, the triangles are congruent.
- **SSS (Side-Side-Side)**: If all three sides of one triangle are equal to the three sides of another triangle, then they are congruent.
- **SAS (Side-Angle-Side)**: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then they are congruent.
3. **Using R as Midpoint**: To show that R is the midpoint of both KN and ML, he could argue based on the information provided, even if not fully clear from the content. If triangle KRM is congruent to triangle NRM (if we assume that KLENM, ZLZM, ZKRL=/NRM give us enough information about angles and sides), it would suggest that R is indeed the midpoint of segments KN and ML due to the properties of congruent triangles (specifically by showing equality of corresponding segments).
4. **Multiple Choice Answer**: Now considering the multiple choices provided:
- Yes, AAS
- Yes, ASA
- Yes, SSS
- Yes, SAS
- No
### Conclusion:
To answer whether the triangles are congruent, and why R must be the midpoint, the most appropriate congruency condition would rely on what angles and sides are definitively equal. If we assume that he can show two angles are equal and a side corresponds (like through AAS or ASA), then the triangles are indeed congruent.
So, to justify R must be the midpoint, Payne would likely argue through triangle congruence and the properties arising from it after establishing angular or side relationships.
Depending on the clarity from provided angles or sides, the specific answer from the options could be AAS or another pair, but the reasoning for the midpoint notion stands firm.
Here's the reasoning:
1. **Identifying Angles and Sides**: According to the problem, we understand Payne's mom has provided some angle relationships and possibly side relationships, represented by KLENM, ZLZM, and ZKRL=/NRM. Although these relationships are not fully spelled out, it suggests that the angles are either equal or have a specific relationship.
2. **Criteria for Triangle Congruence**:
- **ASA (Angle-Side-Angle)**: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
- **AAS (Angle-Angle-Side)**: If two angles and a non-included side of one triangle are equal to two angles and a corresponding side of another triangle, the triangles are congruent.
- **SSS (Side-Side-Side)**: If all three sides of one triangle are equal to the three sides of another triangle, then they are congruent.
- **SAS (Side-Angle-Side)**: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then they are congruent.
3. **Using R as Midpoint**: To show that R is the midpoint of both KN and ML, he could argue based on the information provided, even if not fully clear from the content. If triangle KRM is congruent to triangle NRM (if we assume that KLENM, ZLZM, ZKRL=/NRM give us enough information about angles and sides), it would suggest that R is indeed the midpoint of segments KN and ML due to the properties of congruent triangles (specifically by showing equality of corresponding segments).
4. **Multiple Choice Answer**: Now considering the multiple choices provided:
- Yes, AAS
- Yes, ASA
- Yes, SSS
- Yes, SAS
- No
### Conclusion:
To answer whether the triangles are congruent, and why R must be the midpoint, the most appropriate congruency condition would rely on what angles and sides are definitively equal. If we assume that he can show two angles are equal and a side corresponds (like through AAS or ASA), then the triangles are indeed congruent.
So, to justify R must be the midpoint, Payne would likely argue through triangle congruence and the properties arising from it after establishing angular or side relationships.
Depending on the clarity from provided angles or sides, the specific answer from the options could be AAS or another pair, but the reasoning for the midpoint notion stands firm.