Properties of a Parallelogram, with an Interactive Diagram

Grades 8-9

Question: Draw a parallelogram ABCD.

A parallelogram is a special type of quadrilateral whose opposite sides are parallel and equal in length. In the interactive diagram, you can observe the following fundamental properties that define this figure:

Parallel, Equal Sides: Opposite sides are parallel to each other. Mathematically, this means segment $AB$ is parallel to $DC$ ( $AB \mathrel{/\!/} DC$ ) and segment $AD$ is parallel to $BC$ ( $AD \mathrel{/\!/} BC$ ). Their lengths are also identical: $AB = DC$ and $AD = BC$ .
Consecutive Angles: Angles that share a side (such as the angle at $A$ and the angle at $B$ ) are supplementary. Their sum is always $180^\circ$ : $\begin{align*} \angle A + \angle B = 180^\circ \end{align*}$
Opposite Angles: Angles at opposite vertices are equal to each other ( $\angle A = \angle C$ and $\angle B = \angle D$ ).

To find the coordinates of the fourth vertex

D

so the figure is always a parallelogram, we can use the vector relationship

\vec{AB} = \vec{DC}

, or simply add the vector from

B

C

to point

A

\begin{align*} D_x &= A_x + (C_x - B_x) \\ D_y &= A_y + (C_y - B_y) \end{align*}

In the interactive construction, you can adjust the sliders to change the base length, the side length, and the tilt angle, and watch how these geometric properties are preserved.