Example

Question

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, &L=ZM and ZKRL=/NRM. She asks Payne to justify why R must be the midpoint of both KN and ML. How could Payne do this?
AAS AKLRENMR KRENR
L
M
vertical alwangles
N
MULTIPLE-CHOICE QUESTION
Why is Segment KR congruent to Segment NR?
Reflexive Property
Vertical Angles
CPCTC
AAS

Answer 1

To answer the question of why Segment KR is congruent to Segment NR, the correct choice from the options provided is Reflexive Property.

Here’s how Payne could justify why R must be the midpoint of both KN and ML:

1. Congruent Triangles: Payne's mom provided some information indicating that triangles KLENM and ZKRL are congruent. If these triangles are congruent (e.g., via AAS), then corresponding parts of these triangles must also be congruent.

2. Congruent Segments: By having established that the triangles are congruent, it follows that segments KR and NR, which are corresponding sides in the congruent triangles, must be congruent.

3. Reflexive Property: The Reflexive Property states that a segment (or angle) is congruent to itself. Therefore, if Payne has established that KR is congruent to another segment NR, we can affirm that R must behave symmetrically relative to both segments KN and ML in a way that makes it the midpoint.

To synthesize the complete rationale for why R is the midpoint of KN and ML:

• Since R lies on both KN and ML and the segments KR and NR are proven to be congruent due to the congruency of the triangles (AAS), it follows that R divides both segments in half, confirming it as the midpoint.

So in conclusion:

• For Segment KR is congruent to Segment NR: Reflexive Property (it uses same claim from triangles).
• Why R is the midpoint: Since KR = NR and R divides KN and ML into congruent parts, R must indeed be the midpoint of both.