Download Document

New Jersey Center for Teaching and Learning
Progressive Mathematics Initiative
This material is made freely available at www.njctl.org
and is intended for the non-commercial use of students
and teachers. These materials may not be used for any
commercial purpose without the written permission of
the owners. NJCTL maintains its website for the
convenience of teachers who wish to make their work
available to other teachers, participate in a virtual
professional learning community, and/or provide access
to course materials to parents, students and others.
Click to go to website:
www.njctl.org
7th Grade Math
2D Geometry
2012-01-07
www.njctl.org
Setting the PowerPoint View
Use Normal View for the Interactive Elements
To use the interactive elements in this presentation, do not select the
Slide Show view. Instead, select Normal view and follow these steps to
set the view as large as possible:
• On the View menu, select Normal.
• Close the Slides tab on the left.
• In the upper right corner next to the Help button, click the ^ to minimize
the ribbon at the top of the screen.
• On the View menu, confirm that Ruler is deselected.
• On the View tab, click Fit to Window.
• On the View tab, click Slide Master | Page Setup. Select On-screen
Show (4:3) under Slide sized for and click Close Master View.
• On the Slide Show menu, confirm that Resolution is set to 1024x768.
Use Slide Show View to Administer Assessment Items
To administer the numbered assessment items in this presentation, use
the Slide Show view. (See Slide 11 for an example.)
Table of Contents
Determining if a Triangle is Possible
Special Pairs of Angles
Perimeter & Circumference
Area of Rectangles
Area of Parallelograms
Area of Triangles
Area of Trapezoids
Area of Circles
Mixed Review
Area of Irregular Figures
Area of Shaded Regions
Common Core: 7.G.2, 7.G.4-6, 7.EE.3
Click on a topic to go to that
section
Determining if a
Triangle is Possible
Return to table
of contents
How many different acute triangles can you draw?
How many different right scalene triangles can you draw?
Recall that triangles can be classified according to their
side lengths and the measure of their angles.
Sides:
Scalene - no sides are congruent
Isosceles - two sides are congruent
Equilateral - all three sides are congruent
Angles:
Acute - all three angles are acute
Right - contains one right angle
Obtuse - contains one obtuse angle
There is another property that applies to triangles:
The sum of the lengths of any two sides of a triangle is
greater than the length of the third side.
What does this mean?
If you take the three sides of a triangle and add them in
pairs, the sum is greater than (not equal to) the third side.
If that is not true, then it is not possible to construct a
triangle with the given side lengths.
Example:
Determine if sides of length 5 cm, 8 cm and 12 cm can
form a triangle?
Test all three pairs to see if the sum is greater:
5 + 8 > 12 5 + 12 > 8 8 + 12 > 5
13 > 12
17 > 8 20 > 5
Yes, it is possible to construct a triangle with sides of
lengths 5 cm, 8 cm and 12 cm.
Example:
Determine if sides of length 3 ft, 4 ft and 9 ft can form a
triangle?
Test all three pairs to see if the sum is greater:
3+4>9 3+9>4 4+9>3
7>9
12 > 4
13 > 3
No, it is not possible to construct a triangle with sides of
lengths 3 ft, 4 ft and 9 ft.
Try These:
Determine if triangles can be formed with the following side
lengths:
1. 4 cm, 7 cm, 10 cm
4 + 7 > 10
4 + 10 > 7
7 + 10 > 4
YES
3. 7 ft, 9 ft, 16 ft
7 + 9 = 16
7 + 16 > 9
16 + 9 > 7
NO
2. 24 mm, 20 mm, 30 mm
24 + 20 > 30
24 + 30 > 20
20 + 30 > 24
YES
4. 9 in, 13 in, 24 in
9 + 13 < 24
9 + 24 > 13
13 + 24 > 9
NO
1
Determine if sides of length 5 mm, 14 mm
and 19 mm can form a triangle. Be prepared
to show your work!
A
Yes
B
No
2
Determine if sides of length 6 in, 9 in and 14
in can form a triangle. Be prepared to show
your work!
A
Yes
B
No
3
Determine if sides of length 5 yd, 13 yd and
21 yd can form a triangle. Be prepared to
show your work!
A
Yes
B
No
4
Determine if sides of length 3 ft, 8 ft and
15 ft can form a triangle. Be prepared to
show your work!
A
Yes
B
No
5
Determine if sides of length 5 in, 5 in and
9 in can form a triangle. Be prepared to
show your work!
A
Yes
B
No
6
A triangle could have which of the following
sets of angles?
A
B
C
D
7
A triangle could have which of the following
sets of angles?
A
B
C
D
Example:
Predict the length of the third side of a triangle with sides
of length 12 ft and 16 ft.
Side 1 = 12 ft
Side 2 = 16 ft
The 3rd side must be less than:
12 + 16 > 3rd side
28 ft > 3rd side
The 3rd side must be greater than:
12 + 3rd side > 16
3rd side > 4
The 3rd side must be greater than 4 ft and less than 28 ft.
Example:
Predict the length of the third side of a triangle with sides of
length 9 cm and 15 cm.
Side 1 = 9 cm
Side 2 = 15 cm
The 3rd side must be less than:
9 + 15 > 3rd side
24 cm > 3rd side
The 3rd side must be greater than:
9 + 3rd side > 15
3rd side > 6
The 3rd side must be greater than 6 cm and less than 24 cm.
Try These:
Predict the length of the third side of a triangle whose
known sides are lengths:
1. 13 mm, 20 mm
2. 7 in, 19 in
13 + 20 > Side 3
33 > Side 3
7 + 19 > Side 3
26 > Side 3
13 + Side 3 > 20
Side 3 > 7
7 + Side 3 > 19
Side 3 > 12
7 < side 3 < 33
12 < side 3 < 26
Try These:
Predict the length of the third side of a triangle whose
known sides are lengths:
3. 4 ft, 11 ft
4. 23 cm, 34 cm
4 + 11 > Side 3
15 > Side 3
23 + 34 > Side 3
57 > Side 3
4 + Side 3 > 11
Side 3 > 7
23 + Side 3 > 34
Side 3 > 11
7 < side 3 < 15
11 < side 3 < 57
8
Predict the lower limit of the length of the
third side of a triangle whose known sides
are lengths 6 m and 12 m.
9
Predict the upper limit of the length of the
third side of a triangle whose known sides
are lengths 6 m and 12 m.
10 Predict the lower limit of the length of the
third side of a triangle whose known sides
are lengths 9 in and 17 in.
11
Predict the upper limit of the length of the
third side of a triangle whose known sides
are lengths 9 in and 17 in.
12 Predict the lower limit of the length of the
third side of a triangle whose known sides
are lengths 15 ft and 43 ft.
13 Predict the upper limit of the length of the
third side of a triangle whose known sides
are lengths 15 ft and 43 ft.
Special Pairs
of Angles
Return to table
of contents
Congruent Angles have the same angle
measurement.
14 Are the two angles congruent?
A
Yes
B
No
110
o
75
o
15 Are the two angles congruent angles?
Yes
B
No
40
o
A
40
o
16 Are the two angles congruent?
A
Yes
B
No
105
o
75
o
Complementary Angles are two angles with a sum of 90
degrees.
These two angles are
complementary angles because
their sum is 90.
Notice that they form a right angle
when placed together.
Complementary Angles are two angles with a sum of 90
degrees.
These two angles are
complementary angles because
their sum is 90.
Although they aren't placed
together, they can still be
complementary.
17 What is the measure of A?
A
50
18 What is the measure of A?
575757
57
57
A
57
57
19 Tell whether the two angles are
complementary?
A
Yes
B
No
Angle 1 = 63 degrees
Angle 2 = 27 degrees
20 Tell whether the angles are complementary.
A
Yes
B
No
Angle 1 = 146 degrees
Angle 2 = 44 degrees
Supplementary Angles are two angles with a sum of 180
degrees.
These two angles are
supplementary angles because
their sum is 180.
Notice that they form a straight
angle when placed together.
Supplementary Angles are two angles with a sum
of 180 degrees.
These two angles are
supplementary angles
because their sum is 180.
Although they aren't placed
together, they can still be
supplementary.
21 What is the measurement of angle A?
Angle A
125
o
22 What is the measurement of angle A?
Angle A
40
o
23 Tell whether the two angles are
supplementary.
A
Yes
B
No
Angle 1 = 115 degrees
Angle 2 = 65 degrees
24 Find the supplement of 51
25 Find the complement of 51
26
Find the complement of 27
27 Find the supplement of 27
28 Find the supplement of 102
29 Find the complement of 102
Vertical Angles are two angles that are opposite each
other when two lines intersect.
a
b
d
c
In this example, the vertical angles are:
Vertical angles have the same measurement.
So:
Using what you know about vertical angles, find
the measure of the missing angles.
b
c
a
By Vertical Angles:
By Supplementary Angles:
30 Are angles 2 and 4 vertical angles?
A
B
Yes
No
2
1
3
4
31 Are angles 2 and 3 vertical angles?
A
Yes
B
No
2
3
1
4
32
If angle 1 is 60 degrees, what is the
measure of angle 3? You must be able to
explain why.
2
1
3
4
33 If angle 1 is 60 degrees, what is the
measure of angle 2? You must be able to
explain why.
1
2
3
4
Adjacent Angles are two angles that are next to
each other and have a common ray between them.
This means that they are on the same plane and
they share no internal points.
ABC is adjacent to CBD
A
C
B
How do you know?
·They have a common side (ray CB)
·They have a common vertex (point B)
D
Adjacent or Not Adjacent?
You Decide!
a
b
a
a
b
Adjacent
click to reveal
Not
clickAdjacent
to reveal
b
Not
Adjacent
click
to reveal
34 Which two angles are adjacent to each
other?
A
1 and 4
B
2 and 4
1
4
6
3
2
5
35 Which two angles are adjacent to each
other?
A
B
3 and 6
5 and 4
2
1
5
4
3 6
A transversal is a line that cuts across two or more
(usually parallel) lines.
A
P
E
Q
F
R
A
B
Interactive Activity-Click Here
Corresponding Angles are on the same side of the
transversal and on the same side of the given lines.
In this diagram the
corresponding
angles are:
Transversal
a
c
f
e
g
h
b
d
36 Which are pairs of corresponding angles?
A
B
C
2 and 6
3 and 7
1 and 8
1
3
5
7
6
8
2
4
37 Which are pairs of corresponding angles?
A
B
C
2 and 6
3 and 1
1 and 8
6
2
4
8
1
7
3
5
38 Which are pairs of corresponding angles?
A
B
C
1 and 5
2 and 8
4 and 8
2
1
4
3
5
7
6
8
39 Which pair of angles are not
corresponding?
A
B
C
D
E
5
4
1
8
2
7
6
3
Alternate Exterior Angles are on opposite sides of the
transversal and on the outside of the given lines.
l
In this diagram the
alternate exterior
angles are:
a
c
f
e
g
b
m
d
n
h
Which line is
the transversal?
Alternate Interior Angles are on opposite sides of the
transversal and on the inside of the given lines.
In this diagram the
alternate interior
angles are:
l
a
c
f
e
g
h
b
m
d
n
Same Side Interior Angles are on same sides of the
transversal and on the inside of the given lines.
l
In this diagram the
same side interior
angles are:
a
c
f
e
g
h
b
m
d
n
40
Are angles 2 and 7 alternate exterior angles?
A
Yes
l
B
No
1
5
2
6
4
8
3
m
7
n
41 Are angles 3 and 6 alternate exterior angles?
A
Yes
B
No
l
1
5
2
6
3
m
7
4
n
8
42 Are angles 7 and 4 alternate exterior angles?
A
Yes
B
No
l
1
5
2
3
m
7
4
n
6
8
43 Which angle corresponds to angle 5?
A 3
B 4
l
C 2
D 6
1
3
5
2
6
8
4
m
7
n
44 Which pair of angles are same side interior?
A
B
C
D
3, 4
4, 7
2, 4
6, 1
l
m
1
5
2
6
8
4
3
7
n
45 What type of angles are 3 and 6?
A
B
C
D
E
Alternate Interior Angles
Alternate Exterior Angles
Corresponding Angles
Vertical Angles
Same Side Interior
l
3
1
5
2
6
8
4
m
7
n
46 What type of angles are 5 and 2?
A
B
C
D
E
Alternate Interior Angles
Alternate Exterior Angles
Corresponding Angles
Vertical Angles
Same Side Interior
l
3
1
5
2
6
8
4
m
7
n
47 What type of angles are 5 and 6?
A
B
C
D
E
Alternate Interior Angles
Alternate Exterior Angles
Corresponding Angles
Vertical Angles
Same Side Interior
l
3
1
5
2
6
8
4
m
7
n
48 Are angles 5 and 2 alternate interior angles?
A
Yes
B
No
l
3
1
5
2
6
8
4
m
7
n
49 Are angles 5 and 7 alternate interior angles?
A
Yes
B
No
l
3
1
5
2
6
8
4
m
7
n
50 Are angles 7 and 2 alternate interior angles?
A
Yes
B
No
l
3
1
5
2
6
8
4
m
7
n
51 Are angles 3 and 6 alternate interior angles?
A
Yes
B
No
l
3
1
5
2
6
8
4
m
7
n
Special Case!!!
If parallel lines are cut by a transversal then:
·Corresponding Angles are congruent
·Alternate Interior Angles are congruent
·Alternate Exterior Angles are congruent
n
SO:
1
5
2
6
4
8
3
m
7
l
52 Given the measure of one angle, find the
measures of as many angles as possible.
Which angles are congruent to the given
angle? Type one answer into your
responder.
l
4
5
6
2
1
m
7
8
n
53 Given the measure of one angle, find the
measures of as many angles as possible.
What are the measures of the remaining
angles?
l
4
5
6
2
1
m
7
8
n
54 Given the measure of one angle, find the
measures of as many angles as possible.
Which angles are congruent to the given
angle? Type one of the angles into the
responder.
l
3
1
5
2
8
4
m
7
n
55 Given the measure of one angle, find the
measures of as many angles as possible.
What are the measures of the remaining
angles?
l
3
1
5
2
8
m
7
4
n
Perimeter &
Circumference
Return to table
of contents
Perimeter
Definition: The distance around a two-dimensional figure.
l
w
w
l
Note: (l) represents the Length, or longer side of
the rectangle. (w) represents the Width, or shorter
side of the rectangle. If no units are given, use "u".
Perimeter (P) of a rectangle is found by solving the following
formula:
P = 2l + 2w
Perimeter (P) of a square is found by doing four (4) times
S
Side (s):
P = 4s
Perimeter of a polygon is the sum of the lengths of the
sides.
56 What is the Perimeter (P) of the following
rectangle?
15 ft.
6 ft.
57 What is the Perimeter (P) of the square below?
7
58 What is the Perimeter (P) of the figure?
8 in
59 What is the Perimeter (P) of the figure?
8 cm
10 cm
3 cm
12 cm
Circumference
Definition: The outer boundary of a circle; the
"perimeter" of the circle.
Circumference
Diameter
The circumference (C) of a circle is found by using one of
the following formulas:
C = d
or
C = 2r
or
C = 2r
C = d
or
C = 2r
Diameter (d): Any straight line segment that passes
through the center point of the circle, whose endpoints
are on the circle.
Radius (r): Any line segment from the center point of
the circle, to any point on the circle---radius is 1/2 of the
Diameter. oi of the circle, to any point on the circle---radius
is 1/2 of the Diameter.
C = d
or
C = 2r
Pi (  ), a mathematical constant, is the ratio
of a circle's circumference to its diameter.
Note:
60
What is the Circumference (C) of a circle
with a radius (r) of 7cm? (Use 3.14 for π)
7 cm
61 What is the Circumference (C) of a circle with
a Diameter (D) of 11in.? (Use 3.14 for π)
11 in.
62 Find the circumference of a circle whose
radius is 2.5 meters. (Use 3.14 for π)
63 A circle has diameter 8 yds. What is its
circumference? (Use 3.14 for π)
64 The circumference of a circle is 37.68
cm. What is its radius? (Use 3.14 for π)
Area of Rectangles
Return to table
of contents
2
Area - The number of square units (units ) it takes to
cover the surface of a figure.
2
ALWAYS label units !!!
2
How many 1 ft tiles does it take to cover the rectangle?
Use the squares to find out!
Look for a faster way than covering the whole figure.
12 ft
6 ft
The Area (A) of a rectangle is found by using the
formula:
A = length(width)
A = lw
The Area (A) of a square is found by using the
formula:
A = side(side)
2
A=s
65 What is the Area (A) of the figure?
15 ft
6 ft
66 Find the area of the figure below.
7
67 Dr. Dan wants to keep his kitten from running
through his flower bed by putting up some
fencing. The flower bed is 10 ft. by 6ft. Will
Dr. Dan need to know the area or the
perimeter of his flower bed to keep his kitty
from trampling the flowers?
A
Area
B
Perimeter
68 Now solve the problem....
Dr. Dan wants to keep his kitten from
running through his flower bed by putting
up some fencing. The flower bed is 10 ft. by
6ft. How much fencing will he need?
Area of Parallelograms
Return to table
of contents
Area of a Parallelogram
Let's use the same
process as we did for the rectangle.
2
How many 1 ft tiles fit across the bottom of the
parallelogram?
Area of a Parallelogram.
Let's use the same process as we did for the rectangle.
2
If we build the parallelogram with rows of 14 ft , what
happens?
14 ft
How tall is the
parallelogram?
How can you tell?
How does this help us find the area of the
parallelogram?
5 ft
14 ft
How do you find the area of a
parallelogram?
The Area (A) of a parallelogram is found by using the
formula:
A = base(height)
A = bh
Note: The base & height always form a right angle!
Example.
Find the area of the figure.
4 cm
2.2 cm
1.9 cm
4 cm
click to reveal
2.2 cm
Try These.
11 m
Find the area of the figures.
8
20 m
5
14 m
7
11 m
click to reveal
click to reveal
4
69 Find the area.
11 ft
10 ft
12 ft
70 Find the area.
17 in
12 in
10 in
17 in
12 in
71 Find the area.
7m
13 m
13 m
7m
11 m
72 Find the area.
Area of Triangles
Return to table
of contents
Area of a Triangle
Let's use the same process as2 we did for the rectangle &
parallelogram. How many 1 ft tiles fit across the bottom
of the triangle?
Area of a Triangle
2
If we continue to build the triangle with rows of 10 ft ,
what happens?
10 ft
How tall is the triangle? How can you tell?
How does this help us find the area of the triangle?
4 ft
10 ft
See that the rectangle we built is twice as large as the
triangle. How do you find the area of a triangle?
Find the area of the rectangle, then divide by 2
2
20 ft
Is this true for all triangles?
Let's see!
Calculating base(height) results in 2 triangles!
The Area (A) of a triangle is found by using the formula:
Note: The base & height always form a right angle!
Example.
Find the area of the figure.
4 cm
10 cm
10 cm
click to reveal
6 cm
Try These.
Find the area of the figures.
20
13 ft
9 ft
12 ft
11 ft
14
16
15
click to reveal
click to reveal
73 Find the area.
11 in
8 in
5 in
10 in
74 Find the area
9m
8m
15 m
12 m
Area of Trapezoids
Return to table
of contents
Area of a Trapezoid
·Cut the trapezoid in half horizontally
·Rotate the top half so it lies next to the bottom half
·A parallelogram is created
Base
1
Base
2
Height
See the diagrams below
Base2
Base1
The Area (A) of a trapezoid is found by using the
formula:
Note: The base & height always form a right angle!
Example.
Find the area of the figure.
12 cm
10 cm
11 cm
9 cm
click to reveal
Try These.
Find the area of the figures.
15
13 ft
11 ft
11 ft
9 ft
11 ft
click to reveal
9
7
11
20
click to reveal
75 Find the area of the trapezoid.
4m
6.5 m
10 m
76 Find the area of the trapezoid.
22 cm
8 cm
14 cm
Area of Circles
Return to table
of contents
Area of a Circle
The Area (A) of a Circle is found by solving the following
formula:
Find the
area of the circle.
2
A= πr
7 cm
1. Substitute
the radius into formula.
2
A = π (7)
2. Use 3.14 as an approximation for π.
A = 3.14(49) 2
A = 153.86 cm
3. Don't forget to label the units
as square units.
77 What is the Area (A) of a Circle with a
radius (r) of 8 m?
8m
78 What is the Area (A) of the circle?
79 What is the Area (A) of the circle?
80 A circular sprinkler sprays water with a
radius of 11 ft. How much area can the
sprinkler cover?
81 What is the area of a circle with a diameter
of 24 yds?
82 What is the radius of a circle whose area is
2
254.34 mm ?
2
83 A circular pool has an area of 153.86 ft .
What is its diameter?
Mixed Review:
Perimeter,
Circumference & Area
Return to table
of contents
84 Find the perimeter of the figure.
5 cm
4 cm
3 cm
11 cm
4 cm
85 Find the area of the figure.
8 yd
8 yd
4 yd
9 yd
86 Find the perimeter of the figure.
4m
7m
87 Find the circumference of the figure.
12 in
88 Find the area of the figure.
9 in
5 in
12 in
89 Find the area of the figure.
5 cm
4 cm
3 cm
11 cm
4 cm
90 Find the perimeter of the figure.
9 in
5 in
12 in
91 Find the perimeter of the figure.
8 yd
8 yd
4 yd
9 yd
92
Find the area of the figure.
12 in
93 Find the area of the figure.
4m
7m
94 If you want to place a towel bar 9 3/4 inches
long in the center of a door that is
27 ½ inches wide, how far from the edge of
the door should you put the edge of the bar?
95 A wall is 48" wide. You want to center a
picture frame that is 20" wide on the wall.
How much space will there be between the
edge of the wall and the frame?
Area of
Irregular Figures
Return to table
of contents
Area of Irregular Figures
Method #1
1. Divide the figure into smaller figures
(that you know how to find the area of)
2. Label each small figure and find the area of each
3. Add the areas
4. Label your answer
Example:
Find the area of the figure.
3m
2m
6m
10 m
3m
2m
#2
10 m
#1
6m
Area of Irregular Figures
Method #2
1. Create one large, closed figure.
2. Label the small added figure and find the area.
3. Find the area of the new, large figure
4. Subtract the areas
5. Label your answer
Example:
Find the area of the figure.
3m
2m
6m
10 m
3m
2m
Whole Rectangle
6m
10 m
Extra Rectangle
Try These:
Find the area of each figure.
8 ft
2m
4m
20 ft
2m
5m
10 ft
16 ft
96 Find the area.
Top Rectangle
4'
Bottom Rectangle
2.5'
1.5'
Vertical Rectangle
8.75'
5.25'
2.5'
Total Area
7.75'
Whole New Figure
97 Find the area.
16
New Rectangle
12
19
25
13
35
Total Area
Triangle
98 Find the area.
8 cm
58 cm
15 cm
Rectangle
Total Area
Side Rectangle
99 Find the area.
4 ft.
Bottom Right Rectangle
5 ft.
9 ft.
Half Circle
6 ft.
Total Area
Area of
Shaded
Regions
Return to table
of contents
Area of a Shaded Region
1. Find area of whole figure.
2. Find area of unshaded figure(s).
3. Subtract unshaded area from whole figure.
4. Label answer with units
2
Example
Find the area of the shaded region.
Area Whole Rectangle
20 ft
7 ft
15 ft
Area Unshaded Square
7 ft
Area Shaded Region
Try This
Area Whole Square
Find the area of the shaded region.
Area Circle
Area Shaded Region
14 cm
Try This
Area Trapezoid
Find the area of the shaded region.
20 m
Area Rectangle
3m
12 m
2m
8m
Area Shaded Region
100 Find the area of the shaded region.
Area Whole Rectangle
6'
2'
Area Unshaded
4'
Area Shaded Region
8'
101 Find the area of the shaded region.
Area Parallelogram
11"
8"
Area Triangle
7"
6"
12"
Area Shaded Region
102
Find the area of the shaded region.
Area Whole
8"
14"
8"
Area Rectangle
4"
6"
12"
Area Shaded Region
103 Find the area of the shaded region.
Area Circle
Area Triangle
4 yd
Area Shaded Region
104 A cement path 3 feet wide is poured around a
rectangular pool. If the pool is 15 feet by 7
feet, how much cement was needed to create
the path?
Area Path & Pool
Area Pool
Area Path