Question

mathematics diagnostic assessment
given: ( ab cong dc ) and ( bc cong ad )
prove: ( abcd ) is a parallelogram.
proof:

statements	reasons
2. draw ( ac ) and ( bd ).	2. through any two points, there exists exactly one line
3. ( ac cong ac ) and ( bd cong bd )	3. reflexive property of congruence
4. ( \triangle abc cong \triangle cda ) and ( \triangle bda cong \triangle dbc )	4.
5. ( angle cbd cong angle adb ) and ( angle acd cong angle cab )	5. corresponding parts of congruent triangles are congruent.
6. ( bc parallel ad ) and ( ab parallel dc )	6. alternate interior angles converse
7. ( abcd ) is a parallelogram.	7. a quadrilateral is a parallelogram if opposite sides are parallel.

which reason completes the proof?
a. angle - side - angle congruence
b. corresponding parts of congruent triangles are congruent.
c. side - side - side congruence
d. alternate interior angles

Explanation:

To find the reason for step 4, we look at the prior steps:

1. We know $\overline{AB} \cong \overline{DC}$, $\overline{BC} \cong \overline{AD}$ (given).
2. We established $\overline{AC} \cong \overline{AC}$, $\overline{BD} \cong \overline{BD}$ (reflexive property).
3. For $\triangle ABC \cong \triangle CDA$, we have three pairs of congruent sides: $\overline{AB} \cong \overline{DC}$, $\overline{BC} \cong \overline{AD}$, $\overline{AC} \cong \overline{AC}$. For $\triangle BDA \cong \triangle DBC$, we have $\overline{AD} \cong \overline{BC}$, $\overline{AB} \cong \overline{DC}$, $\overline{BD} \cong \overline{BD}$. This matches the side-side-side (SSS) congruence rule, which states that if all three pairs of corresponding sides of two triangles are congruent, the triangles are congruent.

Answer:

C. side-side-side congruence