# Lesson Title: Properties of Parallelogram

Project AMP Dr. Antonio R. Quesada Director, Project AMP

Lesson Title: Properties of Parallelograms

Lesson Summary: Students will engage in discovering properties of parallelograms by plotting points on a coordinate plane, calculating slopes, measuring angles and calculating distances of side lengths. Additionally, students will make conjectures about the relationship between diagonals in a parallelogram and how they bisect one another.

Key Words:

- parallelogram
- parallel
- diagonals
- midpoint
- slope
- distance formula

**Background Knowledge**:

- plot ordered pairs
- determine slope given two points
- calculate distance of a segment using the distance formula
- determine midpoint of a segment given the endpoints
- relationships between the slopes of parallel and perpendicular segments
- measure angles with a protractor
- using Cabri Jr. Apps in the TI-84 if using the extensions
- angle relationships – supplementary, congruent, consecutive, opposite

**NCTM Standards Addressed**:

Geometric and Spatial Sense Indicator, grade 9, #3:

Analyze two-dimensional figures in a coordinate plane; e.g., use slope and distance formulas to show that a quadrilateral is a parallelogram.

**Learning Objectives**:

- SWBAT discover the following properties of a parallelogram
- Opposite sides parallel
- Opposite sides equal
- Diagonals bisect each other
- Opposite angles congruent
- Consecutive angles supplementary

Materials:

- pencil
- handout
- Geoboard with rubberbands
- protractor
- student lesson sheets
- TI-84 with Cabri Jr. Apps if doing the extensions
- Cabri Jr. Guidebook located at

**Suggested Procedures: The following activities may be modified to fit a jigsaw or activity stations format.**

- Introduction: Find Parallelograms in the Room

Look around the room and try to find at least 5 examples of parallelograms. Write and draw these items in the table on your paper.

- Activity 1: Lengths of Side

Students will plot given points for a parallelogram on the coordinate plane, then calculate distance of the segments using the distance formula. They should then be able to determine that the opposite sides of a parallelogram are equal.

- Activity 2: Slopes

Students will plot given points for a parallelogram on a coordinate plane, then determine the slopes of opposite sides. They should then conclude that the opposite sides have the same slope.

- Activity 3: Diagonal Relationship

Students will plot given points for a parallelogram on a coordinate plane, then use the midpoint formula and the distance formula to determine the point of intersection. They should then conclude that the diagonals bisect each other.

- Activity 4: Angle Relationship

Students will build parallelograms given coordinates and a Geoboard. Students will measure the angles in the constructed parallelogram and discover similarities and differences. They should see that consecutive angles are supplementary and opposite angles are congruent.

- Closure: Re-examine the Parallelograms around the Room

Take out your list from the introduction of this lesson. Are there any objects you would change? How do you know that the objects on the list are parallelograms? Write a sentence next to each describing your thoughts.

- Extension 1: Properties of Parallelograms

- Extension 2: Parallelograms and TI Cabri Jr. Instructions

Assessment:

- Student lesson sheets

- Questioning techniques during inquiry activity

- The Closure Activity

- Quiz

## Properties of Parallelograms

## Worksheet

Lesson Goals: To engage in discovering properties of parallelograms by plotting points on a coordinate plane, calculating slopes, measuring angles and calculating distances of side lengths. To conjecture about the relationship between diagonals in a parallelogram and how they bisect one another.

Directions: Look around the room and try to find at least 5 examples of parallelograms. Write and draw these items in the first and second columns of the table below, titled INTRODUCTION. Please do not write in the third and fourth columns, titled CLOSURE.

### INTRODUCTION

/### CLOSURE

###### Item

/#### Object name

/ Picture /**Is it a parallelogram?**

(yes / no) / Why or

why not?

1

2

3

4

5

Activity One

- On a coordinate grid, plot and label the points A(6,2), B(17,5), C(13,12), and D(2,9).

- What general shape will be formed if you connect A to B, B to C, C to D, and D to A?

Connect each pair of points to verify your answer.

- Use the distance formula, , to determine the length of each of the segments.

Length of =

Length of =

Length of =

Length of =

- What can you conclude about the lengths of the segments in this figure?

Activity Two

- On a coordinate grid, plot and label the points A(-7,5), B(-2,-4), C(8,-8), and D(3,1).

- What general shape will be formed if you connect A to B, B to C, C to D, and D to A?

Connect each pair of points to verify your answer.

- Use the Slope formula, , to determine the slopes of each of the segments.

Slope of =

Slope of =

Slope of =

Slope of =

- What can you conclude about the slopes of the segments in this figure?

Activity Three

- On a coordinate grid, plot and label the points A(-3,0), B(-1,5), C(5,4), and D(3,-1).

- What general shape will be formed if you connect A to B, B to C, C to D, and D to A?

Connect each pair of points to verify your answer.

- Remember that diagonals are segments that connect two non-adjacent vertices of a polygon. How many diagonals will the figure above have? Use a straightedge to draw the diagonals and verify your answer.

- Using the Midpoint formula, , find the midpoint of diagonal . Plot and label this midpoint on the above coordinate plane as point F.

- Use the distance formula, , to determine the length of each of these segments.

Length of =

Length of =

From these lengths, what can you verify?

- Use the distance formula, , to determine the length of each of these segments.

Length of =

Length of =

From these lengths, what can you conclude?

- What can you conclude about the diagonals in this figure?

Activity Four

**Consecutive angles **of a quadrilateral are two angles that share a common ray.

**Opposite angles **of a quadrilateral are two angles that are located on either end of a diagonal, they are directly across from one another.

Consider your Geoboard to be the first quadrant of the coordinate plane, as the picture illustrates below.

Steps:

- Assume that the bottom outside edge of the Geoboard is the x-axis and the left outside edge is the y-axis. Stretch the rubber band around these 4 coordinates to create a parallelogram.

A(1,1)B(3,1)C(3,5)D(5,5)

- Use a protractor to measure the angles at each of the vertices and record them below.

mA =

mB =

mC =

mD =

- Do you notice any similarities or differences among the angle measurements? Explain.

- Now stretch the rubber band around these 4 coordinates to create a different parallelogram.

E(2,1)F(2,4)G(4,2)H(4,5)

- Use a protractor to measure the angles at each of the vertices and record them below

mE =

mF =

mG =

mH =

- Do you notice any similarities or differences among the angle measurements in the second parallelogram? Explain.

- Based on your responses to the #1-6, what general statement(s) can you make about a property or properties found in all parallelograms regarding angle measurements?

- Using your protractor draw a parallelogram below using only the angle measurements to guide you. Label the angle measurements in your parallelogram.

QUIZ

On a coordinate grid, plot and label the points A(-3,3), B(8,5), C(2,-1).

Using the information that you discovered in the previous activities, where could you place point D in order to make a parallelogram?

Please verify your solution by showing the slopes of the sides, the lengths of the sides, the midpoint of the diagonals as well as the angle measures.

After you have found one point that makes a parallelogram, is it possible to find another point that will make a different parallelogram with the three given points? How many different points exist that will form a parallelogram with the three given points? List as many of these points as you can.

Extension 1

I. Constructing a parallelogram

A. What is the definition of a parallelogram?

B. Open the Cabri Jr. APP on your calculator. Construct a parallelogram using the instructions that follow below or use the instructions for Method I or Method II that follow this extension.

1. In F1 open a New window.

2. Using the Segment tool under F2 draw a segment.

3. Label it with the Alp-Num tool in F5.

4. Draw using the Segment tool.

5. With the Parallel tool in F3,6. Then draw a line parallel to

draw a line parallel to through pt. B

through pt. D.

7. Construct the point of 8. With the Quad tool in F2

intersection of the two lines draw quadrilateral ABCD.

and label it C using Intersection

under Point in F2.

9. Hide and using the Hide/Show tool in F5.

C. Grab point A. To do this select point A, press clear (the cursor will switch to a hollow arrow), then press the Alpha key (the cursor will change to a hand), and then move point A around using the arrow keys. Does ABCD appear to stay a parallelogram?

Try grabbing points B, C, and D. What happens?

Which point can you not grab? Why do you think this is so?

II. The Properties

A. Sides

Use **D & Length under the Measure** tool in F5 to measure the four sides.

Then, grab pt. A and change the shape of ABCD. Record the new values.

Change the shape several more times.

What do you notice is true about the lengths of the sides?

B. Angles

Using **Angle in the Measure** tool, find the number of degrees in each angle.

Then, grab a point and change the shape of ABCD. Record the values.

A

B

C

D

Find two relationships that appear to be true about the angles of a parallelogram.

1.

2.

C. Diagonals

Construct diagonals and .

Construct their point of intersection and label it E.

Measure the segments listed below. Then, change the shape and record the

values.

What property about parallelograms can you state?

What is notnecessarily true about the diagonals?

D. Angles formed by the diagonals

1. Measure the angles formed by the intersection of the diagonals. Change

shapes. Can you make any conjecture about the angles?

2. Measure BAE and DAE. Change shapes. Is there any property that seems to be true?

E. List the properties of parallelograms that you have found.

F. Constructions using the properties.

Each person in your group should construct a parallelogram using a different

property.

1

Name: ______Date: ______Period:_____

**Constructing a Parallelogram **

**Using Cabri Jr. - Method I**

1. In the F1 menu highlight New2. Open the F2 window, scroll to

and press ENTER. Segment, and press ENTER.

3. Draw a line segment.4. Highlight Alp-Num in the F5 menu.

5. Label the segment .6. Draw line segment .

7. In F3 select Parallel. 8. Highlight and press ENTER.

9. With the arrow keys move the10. Construct a line parallel to .

parallel line up to pt. D so that D

is blinking and press ENTER.

11. There are three choices under the12. Move the cursor until both lines

**Point tool. Select Intersection**. are blinking. Press ENTER.

Label the point C.

13. Under F2 select the **Quad tool.14. Highlight the Hide/Show** tool in

Select A, then B, C, and D. F5 and press ENTER.

15. Select and press ENTER.16. Hide .

A figure will not disappear until the cursor is moved away from it.

**Constructing a Parallelogram**

**Using Cabri Jr. – Method II**

1. In the F1 menu highlight New2. Open the F2 window, scroll to

and press ENTER. Segment, and press ENTER.

3. Draw a line segment.4. Highlight Alp-Num in the F5 menu.

5. Label the segment .6. Draw line segment .

7. In F3 select Parallel. 8. Highlight and press ENTER.

9. With the arrow keys move the10. We need to copy the length of

parallel line up to pt. D so that onto the line through D. For this

D is blinking and press ENTER. use the Compass tool under F3.

11. Highlight pt. B and press ENTER. 12. Using the arrow keys move the

Then highlight pt. A and press center of the circle from A to D.

ENTER. Be sure D is blinking. ENTER.

Note: When you use the Compass tool, the first point that you highlight will be

on the circle and the second point will be the center. It does not really

matter which one you highlight first since you can move the circle.

13. There are three choices under the14. Move the cursor until both the

**Point tool. Select Intersection**. line and the circle are blinking.

Press ENTER. Label pt. as C.

15. Under F2 select the **Quad tool.16. Highlight Hide/Show** in F5 and

Select A, then B, C, and D. press ENTER.

Note: The order in which you select

the points does not matter.

17. Move the cursor until the circle is 18. Hide line . A figure will not

blinking. Press ENTER. disappear until the cursor is

moved away from it

Properties of Parallelograms

An Inquiry-Based Lesson

Solutions

Activity One

- On a coordinate grid, plot and label the points A(6,2), B(17,5), C(13,12), and D(2,9).

- What general shape will be formed if you connect A to B, B to C, C to D, and D to A?

Connect each pair of points to verify your answer.

Parallelogram (answers may vary)

- Use the distance formula, , to determine the length of each of the segments.

Length of =

Length of =

Length of =

Length of =

- What can you conclude about the lengths of the segments in this figure?

*The lengths of the opposites sides are equal.*

Activity Two

- On a coordinate grid, plot and label the points A(-7,5), B(-2,-4), C(8,-8), and D(3,1).

- What general shape will be formed if you connect A to B, B to C, C to D, and D to A?

Connect each pair of points to verify your answer.

Parallelogram

*Answers may vary, but they should indicate at least a quadrilateral*

- Use the Slope formula, , to determine the slopes of each of the segments.

Slope of =

Slope of =

Slope of =

Slope of =

- What can you conclude about the slopes of the segments in this figure?

*The slopes of opposite sides are the same, therefore, the opposite sides are parallel.*

Activity Three

- On a coordinate grid, plot and label the points A(-3,0), B(-1,5), C(5,4), and D(3,-1).

- What general shape will be formed if you connect A to B, B to C, C to D, and D to A?

Connect each pair of points to verify your answer.

Paralellogram (may vary)

- Remember that diagonals are segments that connect two non-adjacent vertices of a polygon. How many diagonals will the figure above have? Use a straightedge to draw the diagonals and verify your answer.

2 diagonals

- Using the Midpoint formula, , find the midpoint of diagonal . Plot and label this midpoint on the above coordinate plane as point F.

The midpoint is the point (1,2)

- Use the distance formula, , to determine the length of each of these segments.

Length of =

Length of =

From these lengths, what can you verify?* (1,2) is the midpoint of segment AC*

- Use the distance formula, , to determine the length of each of these segments.

Length of =

Length of =

From these lengths, what can you conclude? *(1,2) is also the midpoint of segment BD*

- What can you conclude about the diagonals in this figure?

They bisect each other.

Activity Four

Consecutive angles of a quadrilateral are two angles that share a common ray.

Opposite angles of a quadrilateral are two angles that are located on either end of a diagonal, they are directly across from one another.

Consider your Geoboard to be the first quadrant of the coordinate plane, as the picture illustrates below.

Steps:

- Assume that the bottom outside edge of the Geoboard is the x-axis and the left outside edge is the y-axis. Stretch the rubber band around these 4 coordinates to create a parallelogram.

A(1,1)B(3,1)C(3,5)D(5,5)

- Use a protractor to measure the angles at each of the vertices and record them below.

mA =63

mB =117

mC =63

mD =117

- Do you notice any similarities or differences among the angle measurements? Explain.

*Opposites angles have the same measurements (ex: a=c, b=d)*

*Consecutive angles are supplementary (ex: a+b=180)*

- Now stretch the rubber band around these 4 coordinates to create a different parallelogram.

E(2,1)F(2,4)G(5,2)H(5,5)

- Use a protractor to measure the angles at each of the vertices and record them below

mE =71

mF =109

mG =71

mH =109

- Do you notice any similarities or differences among the angle measurements in the second parallelogram? Explain.

Same as #3

- Based on your responses to the #1-6, what general statement(s) can you make about a property or properties found in all parallelograms regarding angle measurements?

Opposite angles are congruent

*Consecutive angles are supplementary*

- Using your protractor draw a parallelogram below using only the angle measurements to guide you. Label the angle measurements in your parallelogram.

*Answers will vary, look for opposite angles to be congruent and consecutive angles are supplementary.*

QUIZ

On a coordinate grid, plot and label the points A(-3,3), B(8,5), C(2,-1).

Using the information that you discovered in the previous activities, where could you place point D in order to make a parallelogram?

Please verify your solution by showing the slopes of the sides, the lengths of the sides, the midpoint of the diagonals as well as the angle measures.

After you have found one point that makes a parallelogram, is it possible to find another point that will make a different parallelogram with the three given points? How many different points exist that will form a parallelogram with the three given points? List as many of these points as you can.

Slope of AB = 2/11

Slope of BC = 1

Slope of CD must equal 2/11