Question

complete the proof to show that abcd is a parallelogram. bc || ad and cd || ba because the abcd is a parallelogram because both pairs of opposite sides are parallel. therefore, the slope of bc is 4 - 2 / -3 - 2 = 2 / -5 the slope of ad is -4 - (-2) / 3 - (-2) = -4 + 2 / 3 + 2 = -2 / 5

Explanation:

Step1: Recall the property of a parallelogram

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

Step2: Analyze the slopes

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For \(\overline{BC}\), using points \(B(-3,4)\) and \(C(2,2)\), \(m_{BC}=\frac{2 - 4}{2-(-3)}=\frac{-2}{5}\). For \(\overline{AD}\), using points \(A(-2,-2)\) and \(D(3,-4)\), \(m_{AD}=\frac{-4-(-2)}{3-(-2)}=\frac{-2}{5}\). For \(\overline{AB}\), using points \(A(-2,-2)\) and \(B(-3,4)\), \(m_{AB}=\frac{4 - (-2)}{-3-(-2)}=\frac{6}{-1}=-6\). For \(\overline{CD}\), using points \(C(2,2)\) and \(D(3,-4)\), \(m_{CD}=\frac{-4 - 2}{3 - 2}=\frac{-6}{1}=-6\). Since \(m_{BC}=m_{AD}\) and \(m_{AB}=m_{CD}\), both pairs of opposite sides are parallel.

Answer:

slopes of opposite sides are equal