Given the quadrilateral ABCD with vertices A (1,0) , B (5,0), C(7,2), and D (3,2).
Consider the drawing below
To prove that the quadrilateral is a parallelogram it is sufficient to prove that parallel.
Using slope property
The slope of parallel sides are equal
Finding the slope of side AB
Using the given points
[tex]\text{slope AB}=\frac{0-0}{5-1}=0[/tex]Finding the slope of side CD
[tex]\text{Slope CD}=\frac{2-2}{7-3}=0[/tex]Since the slope of side AB = Slope of side CD
Hence, sides AB and CD are parallel.
Using the same steps
[tex]\begin{gathered} \text{Slope AD}=\frac{2-0}{3-1}=\frac{2}{2}=1 \\ \text{Slope BC}=\frac{2-0}{7-5}=\frac{2}{2}=1 \end{gathered}[/tex]Also
Since the slope of side BC = Slope of side AD
Hence, sides BC and AD are parallel.
Therefore, the quadrilateral is a parallelogram.
