Question

given: quadrilateral abcd is a parallelogram.
prove: (overline{ab}congoverline{cd})
(overline{bc}congoverline{ad})
what are the missing statement and reason in step 2 of the proof?
a. statement: (angle baccongangle acd)
reason: alternate interior angles theorem
b. statement: (overline{ab}paralleloverline{cd},overline{bc}paralleloverline{ad})
reason: definition of a parallelogram
c. statement: (\triangle abccong\triangle coa)
reason: sss criterion for congruence
d. statement: (overline{ab}congoverline{cd},overline{bc}congoverline{ad})
reason: given
proof:

statements	reasons
2. ?
3. draw (overline{ac}), a transversal that cuts two pairs of parallel segments.(overline{ab}paralleloverline{cd})(overline{bc}paralleloverline{ad})	drawing a line segment
4. (angle cabcongangle acd)(angle bcacongangle cad)	alternate interior angles theorem
5. (mangle cab = mangle acd)(mangle bca=mangle cad)	definition of congruent
6. (ac = ac)	reflexive property of equality
7. ?	asa criterion for congruence
8. (overline{ab}congoverline{cd})(overline{bc}congoverline{ad})	corresponding sides of congruent triangles are congruent

Explanation:

A parallelogram is defined as a quadrilateral with both pairs of opposite sides parallel. In step 2 of the proof, we need to use the given information (quadrilateral \(ABCD\) is a parallelogram) to state the property that directly follows from the definition of a parallelogram. Option B correctly states \( \overline{AB}\parallel\overline{CD},\overline{BC}\parallel\overline{AD}\) (opposite sides are parallel) with the reason as the definition of a parallelogram. Option A is incorrect because \( \angle BAC\cong\angle ACD\) comes later in the proof (step 4, using the Alternate - Interior Angles Theorem after establishing the parallel sides). Option C is wrong because we need to build up to proving \( \triangle ABC\cong\triangle CDA\) (which would be in step 7 using ASA, not SSS as stated). Option D is incorrect because \( \overline{AB}\cong\overline{CD},\overline{BC}\cong\overline{AD}\) is what we are trying to prove, not something we can state as given in step 2.

Answer:

B. Statement: \( \overline{AB}\parallel\overline{CD},\overline{BC}\parallel\overline{AD}\) Reason: definition of a parallelogram