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# 7.1: Properties of Parallelograms

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Geometry, Chapter 6, Lesson 2.

## Problem 1 – Properties of Parallelograms

In this problem, you will look at the definition of parallelogram and several properties of the parallelograms. Open the Cabri Jr. application by pressing APPS and selecting CabriJr.

1. Define parallelogram.

2. Open the file PAR1 by pressing F1\begin{align*}F1\end{align*}, selecting Open..., and selecting the file. PAR1, shows parallelogram QUAD. Grab and drag point Q\begin{align*}Q\end{align*} to two different positions and record the lengths of the segments in the table (rows 1 and 2). Then, grab and drag point U\begin{align*}U\end{align*} to two different positions and record the data in the table (rows 3 and 4).

Position QU¯¯¯¯¯¯¯¯\begin{align*}\overline{QU}\end{align*} UA¯¯¯¯¯¯¯¯\begin{align*}\overline{UA}\end{align*} AD¯¯¯¯¯¯¯¯\begin{align*}\overline{AD}\end{align*} DQ¯¯¯¯¯¯¯¯\begin{align*}\overline{DQ}\end{align*}
1
2
3
4

3. What do you notice about the lengths of opposite sides of a parallelogram?

Angles of a polygon that share a side are consecutive angles. Angles that do not share a side are called opposite angles.

4. Open the file PAR2, which shows parallelogram QUAD. Grab and drag point Q\begin{align*}Q\end{align*} to four different positions and record the measurement of the angles in the table.

Position Q\begin{align*}\angle{Q}\end{align*} U\begin{align*}\angle{U}\end{align*} A\begin{align*}\angle{A}\end{align*} D\begin{align*}\angle{D}\end{align*}
1
2
3
4

5. What do you notice about consecutive angles of a parallelogram?

6. What do you notice about opposite angles of a parallelogram?

## Problem 2 – Diagonals of Parallelograms

For this problem, you will look at the properties of the diagonals of parallelograms.

7. Open the file PAR3, which shows parallelogram QUAD. Record the lengths of the segments in the table (row 1). Then, grab and drag point U\begin{align*}U\end{align*} to three different positions and record the data in the table (rows 2, 3, and 4).

Position QR¯¯¯¯¯¯¯¯\begin{align*}\overline{QR}\end{align*} AR¯¯¯¯¯¯¯¯\begin{align*}\overline{AR}\end{align*} DR¯¯¯¯¯¯¯¯\begin{align*}\overline{DR}\end{align*} RU¯¯¯¯¯¯¯¯\begin{align*}\overline{RU}\end{align*}
1
2
3
4

8. What do you notice about the diagonals of the parallelogram?

## Problem 3 – Extension: Proving Parallelograms

In this problem, you will explore various properties and see if they guarantee that a quadrilateral is a parallelogram.

9. Does having both pairs of opposite sides congruent guarantee that the quadrilateral is a parallelogram? Draw an example or counterexample.

10. Does having one pair of opposite sides congruent and one pair of opposite sides parallel guarantee that the quadrilateral is a parallelogram? Draw an example or counterexample.

11. Does having one pair of opposite sides parallel and one pair of opposite angles congruent guarantee that the quadrilateral is a parallelogram? Draw an example or counterexample.

## Problem 4 – Extension: Extending the Properties

For this problem,

• Create any quadrilateral and name it GEAR.
• Find the midpoint of each side.
• Connect the midpoints to form a quadrilateral.

12. What type of quadrilateral is formed after you connected the midpoints of GEAR?

13. How can you prove what type of figure is created by connecting the midpoints?

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