Question

complete the proof to show that abcd is a parallelogram.
the slope of \\( \overline { bc } \\) is \\( \frac { 4 - 2 } { - 3 - 2 } = - \frac { 2 } { 5 } \\)
the slope of \\( \overline { ad } \\) is \\( \frac { - 4 - ( - 2 ) } { 3 - ( - 2 ) } = \frac { - 4 + 2 } { 3 + 2 } = - \frac { 2 } { 5 } \\)
\\( \overline { bc } \parallel \overline { ad } \\) and \\( \overline { cd } \parallel \overline { ba } \\) because the
________. therefore,
abcd is a parallelogram because both pairs of opposite
sides are parallel.
\\( \bigcirc \\) lengths of consecutive sides are not equal
\\( \bigcirc \\) slopes of opposite sides are equal
\\( \bigcirc \\) lengths of opposite sides are equal
\\( \bigcirc \\) slopes of consecutive sides are not equal

Explanation:

To determine if two lines are parallel, we use the property that parallel lines have equal slopes. In the context of showing \(ABCD\) is a parallelogram, we check the slopes of opposite sides. The problem already calculated the slopes of \(\overline{BC}\) and \(\overline{AD}\) as equal. If the slopes of \(\overline{CD}\) and \(\overline{BA}\) are also equal (by the same slope - calculation formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\)), then by the definition of a parallelogram (a quadrilateral with both pairs of opposite sides parallel), \(ABCD\) is a parallelogram. The key concept here is that equal slopes of opposite sides imply parallelism.

Answer:

slopes of opposite sides are equal