Given: ABCD is a parallelogram.
Prove: АВС = ACDA
1. ABDC is a parallelogram.
2. AB = DC and BC = DA
3. AC = CA
4. АВС = ACDA
1. Given
2. Opposite sides of a parallelogram are congruent.
3. _____
4. _____
A. • Transitive Property of =; SAS
B. • Reflexive Property of =; SAS
C. • Transitive Property of =; SSS
D. • Reflexive Property of ⅚; SSS
To prove that triangles ABC and ACDA are congruent, we need to fill in the blanks for steps 3 and 4 by identifying the correct properties and congruence criteria used in these steps.
From step 2, we have that AB = CD (because they are opposite sides of a parallelogram) and BC = DA (also because they are opposite sides of a parallelogram).
Step 3 should involve the segment AC, which is a diagonal of the parallelogram, and is being used in both triangles ABC and ACDA. Since a segment is always equal to itself in length, we can say that AC is equal to itself by the Reflexive Property of Equality.
3. Reflexive Property of Equality (∵ segment AC equals itself)
In step 4, we are asserting that triangle ABC is congruent to triangle CDA. We have three pairs of congruent sides now (AB = CD, BC = DA, and AC = AC), satisfying the criteria for Side-Side-Side (SSS) congruence postulate, which states that if all three sides of one triangle are congruent to all three sides of another triangle, then the triangles are congruent.
4. Triangle ABC is congruent to triangle CDA by the SSS (Side-Side-Side) Congruence Postulate.
Therefore, the correct answer to complete the proof would be:
3. Reflexive Property of =;
4. SSS
The appropriate answer is thus D, although there is a typo “Reflexive Property of ⅚,” but it should correctly be, “Reflexive Property of =; SSS”.