Provide the reasons for the following proof.
The figure shows triangle J K N and triangle L K M formed by intersecting segments J M and N L at point K. Segment J K is marked with 2 hash marks. Segment K L is marked with 2 hash marks. Segment N K is marked with 1 hash mark. Segment K M is marked with 1 hash mark.
Given: segment J K is congruent to segment K L and segment N K is congruent to segment K M
Prove: triangle N K J is congruent to triangle M L K
Statements Reasons
1. segment J K is congruent to segment K L and segment N K is congruent to segment K M 1. Given.
2. angle J K N is congruent to angle L K M 2. ________________
3. triangle N K J is congruent to triangle M L K
3. ________________
(1 point)
Responses
Reflexive Property of congruent to; SAS
Reflexive Property of Image with alt text: congruent to ; SAS
Reflexive Property of congruent to; ASA
Reflexive Property of Image with alt text: congruent to ; ASA
Vertical angles are congruent; ASA
Vertical angles are congruent; ASA
Vertical angles are congruent; SAS
The reasons for the proof are:
2. Vertical angles are congruent (by the Vertical Angle Theorem)
- This is because angle JKN and angle LKM are formed by intersecting lines, which creates vertical angles.
3. ASA (Angle-Side-Angle) congruence theorem
- This is because we have angle JKN congruent to angle LKM (by reason 2)
and side JK congruent to side KL, as well as side KN congruent to side KM (given in the problem statement).
Therefore, triangle NKJ is congruent to triangle MLK (by ASA congruence theorem).