7.1: Properties of Parallelograms
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
- This activity was written to be explored with the Cabri Jr. app on the TI-84.
- Before beginning this activity, make sure that all students have the Cabri Jr. application, and the Cabri Jr. files PAR1.8xv, PAR2.8xv, PAR3.8xv loaded on their TI-84 calculators.
- In Cabri Jr., to grab a point hover the cursor over the point and press ALPHA. To release press ALPHA or ENTER. To move a point after grabbing it, use the arrow keys.
- To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
- To download the calculator files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=11932 and select PAR, PAR2, and PAR3.
Associated Materials
- Student Worksheet: Properties of Parallelograms http://www.ck12.org/flexr/chapter/9691
- Cabri Jr. Application
- PAR.8xv, PAR2.8xv, and PAR3.8xv
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:
- both pairs of opposite sides are congruent
- both pairs of opposite angles are congruent
- both pairs of opposite sides are parallel
- one pair of opposite sides is both parallel and congruent
- 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.
2. Sample answers:
Position | \begin{align*}\overline{QU}\end{align*} | \begin{align*}\overline{UA}\end{align*} | \begin{align*}\overline{AD}\end{align*} | \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.
4. Sample answers:
Position | \begin{align*}\angle{Q}\end{align*} | \begin{align*}\angle{U}\end{align*} | \begin{align*}\angle{A}\end{align*} | \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.
7. Sample answers:
Position | \begin{align*}\overline{QR}\end{align*} | \begin{align*}\overline{AR}\end{align*} | \begin{align*}\overline{DR}\end{align*} | \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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