Download Properties of Parallelograms and Rectangles

Properties of Parallelograms and Rectangles
Step 1: Go to http://standards.nctm.org/document/eexamples/chap5/5.3/index.htm
or search “dynamic rectangle” and it is the second result which starts with Example
5.3
Step 2: Click the “stand-alone applet” and play around with the parallelogram and
rectangle.
Both shapes will maintain certain features or properties and it is your job to figure out
what some of them are through this activity.
Step 3: Answer the following questions:
1. What is alike about all the figures produced by the dynamic rectangle?
2. What is alike about all the figures produced by the dynamic parallelogram?
3. What common characteristics do parallelograms and rectangles share?
Step 4: Look at the figures under “Other tasks” and place your applet on the screen where
you can view both the applet and the figures.
Before you try to make the shapes, take a guess at which shapes the parallelogram and
rectangle can make by placing a check mark by the ones you think will work and an X by
the ones you don’t think will work in the chart.
Figures
A
B
C
D
E
Guess
Guess
Result
Result
Parallelogram
Rectangle
Parallelogram
Rectangle
Step 5: Now check your predictions by making the shape of the different figures with your
applet. Mark your results in the chart under the two result columns.
Step 6: Answer the following questions.
1. Can the dynamic rectangle make all the shapes that the dynamic parallelogram
can make? Can the dynamic parallelogram make all the shapes that the dynamic
rectangle can make?
2. Describe how to decide if the dynamic rectangle can make a particular shape.
3. Describe how to decide if the dynamic parallelogram can make a particular
shape.
Definition of Parallelogram:
Stop
Discuss results in class.
op
Step 1: Go to GSP.
Step 2: Create 3 parallelograms with different shapes and sizes and label them. (I will
demonstrate how to do so)
Step 3: Measure angles and segments of each of these.
Step 4: Make as many conjectures as you can come up with about parallelograms based on
the relationships between segments and angles that all three parallelograms have.
Write them down on the next page.
Conjectures:
Step 5: Draw diagonals on each of the parallelograms (the lines that connect the opposite
points/angles)
Step 6: Measure the different lengths of the diagonals. Notice how the two diagonals of
each parallelogram intersect each other. Construct an intersection and connect the
angles and intersection by a segment. Measure the lengths of the, now, four
segments of each parallelogram. (I will demonstrate this).
Step 7: Make a conjecture concerning the diagonals of parallelograms from the information
you have collected from your three examples.
Conjecture:
Stop
op
Discuss results in class.
Find the measure of <GFB.
mBAC = 143
4 cm
B
A
G
1 cm
segment
Find the measure of <ACF.
v
F
Find the value of z.
C
Find the length of
AG.
mGFC = 49
Homework/Class work: pg. 415, numbers 16-33, 52-55, 61-63
Properties of Parallelograms and Rectangles
(Teacher Edition)
Step 1: Go to http://standards.nctm.org/document/eexamples/chap5/5.3/index.htm
or search “dynamic rectangle” and it is the second result which starts with Example
5.3
Step 2: Click the “stand-alone applet” and play around with the parallelogram and
rectangle.
Both shapes will maintain certain features or properties and it is your job to figure out
what some of them are through this activity.
Step 3: Answer the following questions:
1. What is alike about all the figures produced by the dynamic rectangle?
They have 90 degree angles and have parallel opposite sides
2. What is alike about all the figures produced by the dynamic parallelogram?
They have parallel opposite sides
3. What common characteristics do parallelograms and rectangles share?
They have parallel opposite sides
Step 4: Look at the figures under “Other tasks” and place your applet on the screen where
you can view both the applet and the figures.
Before you try to make the shapes, take a guess at which shapes the parallelogram and
rectangle can make by placing a check mark by the ones you think will work and an X by
the ones you don’t think will work in the chart.
Figures
A
B
C
D
E
Guess
Guess
Result
Result
Parallelogram
Rectangle
Parallelogram
√
√
√
√
X
Rectangle
√
√
√
X
X
Step 5: Now check your predictions by making the shape of the different figures with your
applet. Mark your results in the chart under the two result columns.
Step 6: Answer the following questions.
1. Can the dynamic rectangle make all the shapes that the dynamic parallelogram
can make? Can the dynamic parallelogram make all the shapes that the dynamic
rectangle can make?
Dynamic rectangle cannot make all the shapes as the parallelogram but the
parallelogram can make all the ones the rectangle can make
2. Describe how to decide if the dynamic rectangle can make a particular shape.
If the shape has right angles
3. Describe how to decide if the dynamic parallelogram can make a particular
shape.
If the shape has parallel lines
Definition of Parallelogram: Quadrilateral with parallel opposite sides
Stop
Discuss results in class.
op
 When you tried to make the figures were any of you surprised with your
results?
 Did any of you guess something different than what you found as a result?
 What did you notice about the flexibility of the rectangle and the
parallelogram? Which one had more restrictions?
 [Ask Second set of questions.
]
 Give me some ideas for a definition.
o
Are they all 4 sided?
o
What is a name for a 4 sided polygon?
Step 1: Go to GSP.
Step 2: Create 3 parallelograms with different shapes and sizes and label them. (I will
demonstrate how to do so) [Demonstrate]
Step 3: Measure angles and segments of each of these. [Demonstrate]
Step 4: Make as many conjectures as you can come up with about parallelograms based on
the relationships between segments and angles that all three parallelograms have. Write
them down on the next page.
Conjectures:
Theorem 8.3: Opp. Sides of parallelograms are congruent.
Theorem 8.4: Opp. Angles of parallelograms are congruent.
o What do you notice about angles that are next to each other?
o If you add them up what do they equal?
o Do we have a term for angles that add up to that number?
Theorem 8.5: Consecutive angles in parallelograms are supplementary
(=180)
o What would happen if one of the angles had a right angle?
o Demonstrate…If I try making one into a right angle what happened to
all my other angles?
Theorem 8.6: If a parallelogram has one right angle then it has four right
angles.
Step 5: Draw diagonals on each of the parallelograms (the lines that connect the opposite
points/angles)
Step 6: Measure the different lengths of the diagonals. Notice how the two diagonals of
each parallelogram intersect each other. Construct an intersection and connect the
angles and intersection by a segment. Measure the lengths of the, now, four
segments of each parallelogram. (I will demonstrate this). [Demonstrate]
Step 7: Make a conjecture concerning the diagonals of parallelograms from the information
you have collected from your three examples.
Conjecture:
Theorem 8.7: Diagonals of a parallelogram bisect each other.
o So what does bisect mean? [divides into equal parts]
Stop
op
Discuss results in class.
[Do on board]
mBAC = 143
4 cm
B
Find the measure of <ACF.
A
G
1 cm
v
F
mGFC = 49
Find the measure of <GFB.
Find the value of z.
C
Find the length of segment AG.
Homework/Class work: pg. 415, numbers 16-33, 52-55, 61-63