Justify the last two steps of the proof.
Given: ABCD is a parallelogram.
Prove: ΔABC is congruent to ΔCDA
The figure shows parallelogram A B C D with diagonal A C.
ABDC is a parallelogram.
AB is congruent to DC and BC is congruent to DA
AC is congruent to CA
ΔABC is congruent to ΔCDA
Given
Opposite sides of a parallelogram are congruent.
_____________________?
_____________________?
(1 point)
Responses
Transitive Property of congruent; SAS
Transitive Property of Image with alt text: congruent ; SAS
Reflexive Property of congruent; SAS
Reflexive Property of Image with alt text: congruent ; SAS
Transitive Property of congruent; SSS
Transitive Property of Image with alt text: congruent ; SSS
Reflexive Property of congruent; SSS
Reflexive Property of Image with alt text: congruent ; SSS
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The last two steps of the proof are:
(1) Transitive Property of congruence; SAS (Side-Angle-Side): This step is justified because we are given that AB is congruent to DC and BC is congruent to DA. In addition, we know that AC is congruent to CA (Reflexive Property of congruence) and the included angle BAC is congruent to the included angle CAD (Alternate Interior Angles Theorem). Therefore, by the SAS congruence criterion, we can conclude that ΔABC is congruent to ΔCDA.
(2) Transitive Property of congruence; SSS (Side-Side-Side): This step is justified because we are given that AB is congruent to DC, BC is congruent to DA, and AC is congruent to CA. Therefore, by the SSS congruence criterion, we can conclude that ΔABC is congruent to ΔCDA.
The last two steps of the proof can be justified using the transitive property of congruence and the SAS (side-angle-side) congruence criterion.
By the transitive property of congruence, if AB is congruent to DC and BC is congruent to DA, then ABDC is a parallelogram.
Using the SAS congruence criterion, if ABDC is a parallelogram and AC is congruent to CA, then we can conclude that triangle ABC is congruent to triangle CDA.
To justify the last two steps of the proof, we need to use the congruence postulate of SAS (Side-Angle-Side).
In this case, we know that AB is congruent to DC and BC is congruent to DA from the given information. We also know that AC is congruent to itself by the reflexive property of congruence.
By using the transitive property of congruence, we can conclude that ΔABC is congruent to ΔCDA.