Question

quadrilateral abcd is shown on the coordinate plane.
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 da × slope of ab = -1
b. slope of ab = slope of bc and slope of cd = slope of da
c. side length of ab = side length of cd and slope of bc × slope of cd = -1
d. slope of ab = slope of cd and slope of bc = slope of da

Explanation:

Step1: Recall the property of a parallelogram

A quadrilateral is a parallelogram if both pairs of opposite sides are parallel.

Step2: Use the slope formula

The slope of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For a quadrilateral \(ABCD\) to be a parallelogram, \(\overline{AB}\parallel\overline{CD}\) and \(\overline{BC}\parallel\overline{DA}\). If two lines are parallel, their slopes are equal. So we need to show that slope of \(\overline{AB}=\) slope of \(\overline{CD}\) and slope of \(\overline{BC}=\) slope of \(\overline{DA}\)

Answer:

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