Question

mathematics diagnostic assessment
given: ab ≅ dc and bc ≅ ad
prove: abcd is a parallelogram.
proof:
statements reasons

1. ab ≅ dc and bc ≅ ad 1. given
2. draw ac and bd. 2. through any two points, there exists exactly one line.
3. ac ≅ ac and bd ≅ bd 3. reflexive property of congruence
4. △abc ≅ △cda and △bda ≅ △dbc 4.
5. ∠cbd ≅ ∠adb and ∠acd ≅ ∠cab 5. corresponding parts of congruent triangles are congruent
6. bc || ad and ab || 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. alternate interior angles
b. corresponding parts of congruent triangles are congruent.
c. angle - side - angle congruence
d. side - side - side congruence

Explanation:

We know that AB ≅ DC and BC ≅ AD (given). When we draw AC and BD, AC ≅ AC and BD ≅ BD by the reflexive property of congruence. To prove △ABC ≅ △CDA and △BDA ≅ △DBC, since we have three pairs of corresponding sides congruent (AB ≅ DC, BC ≅ AD, AC ≅ AC for △ABC and △CDA; AB ≅ DC, AD ≅ BC, BD ≅ BD for △BDA and △DBC), we use the side - side - side (SSS) congruence criterion.

Answer:

D. side - side - side congruence