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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.

ID: 11932

Time Required: 15 minutes

## Activity Overview

Students will explore the various properties of parallelograms. As an extension, students can also explore necessary and sufficient conditions that guarantee that a quadrilateral is a parallelogram.

Topic: Quadrilaterals & General Polygons

• Inductive Reasoning
• Parallelograms

Teacher Preparation and Notes

Associated Materials

## Problem 1 – Properties of Parallelograms

Students will begin this activity by looking at properties of parallelograms. They will discover that opposite sides are congruent, opposite angles are congruent, and that consecutive angles are supplementary.

As an extension, students can prove each of these using parallel lines and transversals. Students will need to know the properties of alternate interior angles, same-side interior angles, and corresponding angles.

## Problem 2 – Diagonals of Parallelograms

In Problem 2, students are asked to investigate the diagonals of a parallelogram. Students should discover that the diagonals of a parallelogram bisect each other. This particular wording may be hard for students discover independently.

## Problem 3 – Extension: Proving Parallelograms

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

Students should know that the following prove that a quadrilateral is a parallelogram:

1. both pairs of opposite sides are congruent
2. both pairs of opposite angles are congruent
3. both pairs of opposite sides are parallel
4. one pair of opposite sides is both parallel and congruent
5. the diagonals bisect each other

## Problem 4 – Extension: Extending the Properties

For this problem, students will create any quadrilateral and name it GEAR. Next, students will find the midpoint of each side and connect the midpoints to form a quadrilateral. Students will use the properties of a parallelogram to see that a parallelogram is always created.

## Solutions

1. A quadrilateral with both pairs of opposite sides parallel.

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 4.50 2.44 4.50 2.44
2 4.50 2.94 4.50 2.94
3 4.90 2.94 4.90 2.94
4 5.50 2.94 5.50 2.94

3. The opposite sides are congruent.

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*}
1 100 80 100 80
2 106 74 106 74
3 124 56 124 56
4 77 103 77 103

5. The consecutive angles are supplementary.

6. The opposite angles are congruent.

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 2.84 2.84 3.29 3.29
2 2.94 2.94 3.45 3.45
3 3.06 3.06 3.58 3.58
4 3.19 3.19 3.71 3.71

8. The diagonals bisect each other.

9. Yes

10. No

11. Yes

12. Parallelogram

13. Sample: Both pairs of opposite sides are parallel.

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