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:
1.
2.
3.
4.
5.
6.
parallelogram
parallel
diagonals
midpoint
slope
distance formula
Background Knowledge:
1.
2.
3.
4.
5.
6.
7.
8.
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:
1. SWBAT discover the following properties of a parallelogram
a. Opposite sides parallel
b. Opposite sides equal
c. Diagonals bisect each other
d. Opposite angles congruent
e. Consecutive angles supplementary
Materials:
1.
2.
3.
4.
5.
6.
7.
pencil
handout
Geoboard with rubberbands
protractor
student lesson sheets
TI-84 with Cabri Jr. Apps if doing the extensions
Cabri Jr. Guidebook located at
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Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
http://education.ti.com/guidebooks/apps/83cabri_jr/ti83pcabrijr_eng.pdf
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Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Suggested Procedures: The following activities may be modified to fit a
jigsaw or activity stations format.
1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. Extension 1: Properties of Parallelograms
8. Extension 2: Parallelograms and TI Cabri Jr. Instructions
Assessment:
1. Student lesson sheets
2. Questioning techniques during inquiry activity
3. The Closure Activity
4. Quiz
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Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
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
Item
Object name
CLOSURE
Is it a
parallelogram?
(yes / no)
Picture
1
2
3
4
5
4
Why or
why not?
Project AMP
Dr. Antonio R. Quesada
5
Director, Project AMP
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Activity One
1. On a coordinate grid, plot and label the points A(6,2), B(17,5), C(13,12), and D(2,9).
2. 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.
3. Use the distance formula, d  ( x1  x2 )2  ( y1  y2 )2 , to determine the length of each
of the segments.
Length of AB
=
Length of BC
=
Length of CD
=
Length of DA
=
4. What can you conclude about the lengths of the segments in this figure?
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Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Activity Two
1. On a coordinate grid, plot and label the points A(-7,5), B(-2,-4), C(8,-8), and D(3,1).
2. 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.
3. Use the Slope formula, m 
( y1  y2 )
, to determine the slopes of each of the segments.
( x1  x2 )
Slope of AB
=
Slope of BC
=
Slope of CD
=
Slope of DA
=
4. What can you conclude about the slopes of the segments in this figure?
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Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Activity Three
1. On a coordinate grid, plot and label the points A(-3,0), B(-1,5), C(5,4), and D(3,-1).
2. 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.
3. 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.
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Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
 x  x   y  y 
4. Using the Midpoint formula, midpoint   1 2 , 1 2  , find the midpoint of
2
2


diagonal AC . Plot and label this midpoint on the above coordinate plane as point F.
5. Use the distance formula, d  ( x1  x2 )2  ( y1  y2 )2 , to determine the length of each
of these segments.
Length of AF
=
Length of CF
=
From these lengths, what can you verify?
6. Use the distance formula, d  ( x1  x2 )2  ( y1  y2 )2 , to determine the length of each
of these segments.
Length of BF
=
Length of DF
=
From these lengths, what can you conclude?
7. What can you conclude about the diagonals in this figure?
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Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
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.
y-axis
x-axis
Steps:
1. 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)
2. Use a protractor to measure the angles at each of the vertices and record them below.
m A =
m B =
m C =
m D =
3. Do you notice any similarities or differences among the angle measurements? Explain.
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Project AMP
Dr. Antonio R. Quesada
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Director, Project AMP
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
4. 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)
5. Use a protractor to measure the angles at each of the vertices and record them below
m E =
m F =
m G =
m H =
6. Do you notice any similarities or differences among the angle measurements in the
second parallelogram? Explain.
7. 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?
8. Using your protractor draw a parallelogram below using only the angle measurements
to guide you. Label the angle measurements in your parallelogram.
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Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
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.
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Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
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 AB with the Alp-Num tool in F5.
4. Draw AD using the Segment tool.
5. With the Parallel tool in F3,
draw a line parallel to AB
through pt. D.
6. Then draw a line parallel to
AD through pt. B
7. Construct the point of
intersection of the two lines
and label it C using Intersection
under Point in F2.
8. With the Quad tool in F2
draw quadrilateral ABCD.
9. Hide BC and DC using the Hide/Show tool in F5.
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Project AMP
Dr. Antonio R. Quesada
15
Director, Project AMP
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
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.
Length in
1st shape
Side
Length in
2nd shape
AB
BC
DC
AD
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.
Angle
A
B
C
D
1st shape
2nd shape
Find two relationships that appear to be true about the angles of a
parallelogram.
1.
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2.
Dr. Antonio R. Quesada
Director, Project AMP
C. Diagonals
Construct diagonals AC and BD .
Construct their point of intersection and label it E.
Measure the segments listed below. Then, change the shape and record the
values.
Length
1st shape
2nd shape
AC
BD
AE
BE
CE
DE
What property about parallelograms can you state?
What is not necessarily 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.
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Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
F. Constructions using the properties.
Each person in your group should construct a parallelogram using a different
property.
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Name: ______________________________________ Date: __________ Period:_____
Constructing a Parallelogram
Using Cabri Jr. - Method I
1. In the F1 menu highlight New
and press ENTER.
2. Open the F2 window, scroll to
Segment, and press ENTER.
3. Draw a line segment.
4. Highlight Alp-Num in the F5 menu.
5. Label the segment AB .
6. Draw line segment AD .
7. In F3 select Parallel.
8. Highlight AB and press ENTER.
9. With the arrow keys move the
parallel line up to pt. D so that D
is blinking and press ENTER.
10. Construct a line parallel to AD .
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Name: ______________________________________ Date: __________ Period:_____
11. There are three choices under the
Point tool. Select Intersection.
12. Move the cursor until both lines
are blinking. Press ENTER.
Label the point C.
13. Under F2 select the Quad tool.
Select A, then B, C, and D.
14. Highlight the Hide/Show tool in
F5 and press ENTER.
15. Select BC and press ENTER.
16. Hide DC .
A figure will not disappear until the cursor is moved away from it.
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Name: ______________________________________ Date: __________ Period:_____
Constructing a Parallelogram
Using Cabri Jr. – Method II
1. In the F1 menu highlight New
and press ENTER.
2. Open the F2 window, scroll to
Segment, and press ENTER.
3. Draw a line segment.
4. Highlight Alp-Num in the F5 menu.
5. Label the segment AB .
6. Draw line segment AD .
7. In F3 select Parallel.
8. Highlight AB and press ENTER.
9. With the arrow keys move the
parallel line up to pt. D so that
D is blinking and press ENTER.
10. We need to copy the length of AB
onto the line through D. For this
use the Compass tool under F3.
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Name: ______________________________________ Date: __________ Period:_____
11. Highlight pt. B and press ENTER.
Then highlight pt. A and press
ENTER.
12. Using the arrow keys move the
center of the circle from A to D.
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 the
Point tool. Select Intersection.
14. Move the cursor until both the
line and the circle are blinking.
Press ENTER. Label pt. as C.
15. Under F2 select the Quad tool.
Select A, then B, C, and D.
16. Highlight Hide/Show in F5 and
press ENTER.
Note: The order in which you select
the points does not matter.
17. Move the cursor until the circle is
blinking. Press ENTER.
18. Hide line DC . A figure will not
disappear until the cursor is
moved away from it
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