Question

given: quadrilateral abcd is a parallelogram.
prove: ( overline{ab} cong overline{cd} )( overline{bc} cong overline{ad} )

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what are the missing statement and reason in step 2 of the proof?
a. statement: ( angle bac cong angle acd )
reason: alternate interior angles theorem
b. statement: ( overline{ab} cong overline{cd}, overline{bc} cong overline{ad} )
reason: given
c. statement: ( overline{ab} parallel overline{cd}, overline{bc} parallel overline{ad} )
reason: definition of a parallelogram
d. statement: ( \triangle abc cong \triangle cda )
reason: sss criterion for congruence
e. statement: ( angle abc cong angle bcd, angle bad cong angle adc )
reason: for parallel lines cut by a transversal, corresponding angles are congruent

Explanation:

To determine the missing statement and reason in step 2, we analyze the proof structure. The first step states that \(ABCD\) is a parallelogram (given). The definition of a parallelogram is a quadrilateral with both pairs of opposite sides parallel. So, the statement should be \(\overline{AB} \parallel \overline{CD}\) and \(\overline{BC} \parallel \overline{AD}\), with the reason being the definition of a parallelogram.

• Option A: The angles \(\angle BAC \cong \angle ACD\) would come later (after identifying parallel sides and using alternate interior angles), so this is not step 2.
• Option B: The statement \(\overline{AB} \cong \overline{CD}, \overline{BC} \cong \overline{AD}\) is what we are trying to prove, not given in step 2.
• Option C: Matches the definition of a parallelogram (opposite sides parallel) as the statement and reason.
• Option D: Triangle congruence (\(\triangle ABC \cong \triangle CDA\)) happens later in the proof, not step 2.
• Option E: The angles \(\angle ABC \cong \angle BCD\) etc. are consecutive angles, and the reason about corresponding angles is incorrect (they are same - side interior angles, supplementary, not congruent in general for parallelograms).

Answer:

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