Question

what needs to be proven to conclude that quadrilateral abcd is a parallelogram?
a. side length of bc = side length of da and slope of \\(\overline{da}\\) × slope of \\(\overline{ab}\\) = -1
b. side length of ab = side length of cd and slope of \\(\overline{bc}\\) × slope of \\(\overline{cd}\\) = -1
c. slope of \\(\overline{ab}\\) = slope of \\(\overline{cd}\\) and slope of \\(\overline{bc}\\) = slope of \\(\overline{da}\\)
d. slope of \\(\overline{ab}\\) = slope of \\(\overline{bc}\\) and slope of \\(\overline{cd}\\) = slope of \\(\overline{da}\\)

Explanation:

To conclude a quadrilateral is a parallelogram, we use the definition: a quadrilateral with both pairs of opposite sides parallel. Parallel lines have equal slopes. So we need to prove the slope of \(\overline{AB}\) equals the slope of \(\overline{CD}\) (opposite sides \(AB\) and \(CD\) parallel) and the slope of \(\overline{BC}\) equals the slope of \(\overline{DA}\) (opposite sides \(BC\) and \(DA\) parallel).

• Option A: Slope product \(-1\) implies perpendicular, not parallel. Incorrect.
• Option B: Slope product \(-1\) implies perpendicular, not parallel. Incorrect.
• Option C: Matches the requirement for opposite sides to be parallel (equal slopes). Correct.
• Option D: These are adjacent sides, not opposite. Incorrect.

Answer:

C. slope of \(\overline{AB}\) = slope of \(\overline{CD}\) and slope of \(\overline{BC}\) = slope of \(\overline{DA}\)