2D_Geometry_Presentationabridged.notebook
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Table of Contents
Table of Contents
Click on a topic to go to that section
Special Pairs of AnglesPerimeter & Circumference
Determining if a Triangle is Possible
Area of Rectangles
Area of Irregular Figures Area of Shaded Regions
Common Core: 7.G.2, 7.G.46, 7.EE.3
Area of ParallelogramsArea of TrianglesArea of TrapezoidsArea of CirclesMixed Review
May 271:59 PM
Determining if a Triangle is Possible
Jan 281:39 PM
Recall that triangles can be classified according to their side lengths and the measure of their angles.
Sides:Scalene no sides are congruentIsosceles two sides are congruentEquilateral all three sides are congruent
Angles:Acute all three angles are acuteRight contains one right angleObtuse contains one obtuse angle
Jan 281:39 PM
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.
Jan 281:39 PM
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.
Jan 281:39 PM
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.
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Jan 282:19 PM
1A triangle could have which of the following sets of angles?
A
B
C
D
Pull
Pull B
Jan 282:19 PM
2A triangle could have which of the following sets of angles?
A
B
C
D
Pull
Pull A, D
May 271:59 PM
Special Pairsof Angles
Return to Table of Contents
congruent angles
Congruent Angles have the same angle measurement.
Complementary Angles
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.
Pull
Pull 40 + 50 = 90
Complementary Angles
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.
Pull
Pull
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May 52:36 PM
3What is the measure of A?
50
Pull
Pull
Supplementary Angles
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.
Pull
Pull 40+140=180
Supplementary Angles
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.
Pull
Pull 110+70=180
May 53:18 PM
4What is the measurement of angle A?
Angle A125o
Pull
Pull
May 53:20 PM
5What is the measurement of angle A?
Angle A 40o
Pull
Pull
May 62:07 PM
6Tell whether the two angles are supplementary.
Yes No
Angle 1 = 115 degrees Angle 2 = 65 degrees
Pull
Pull
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Vertical Angles
Vertical Angles are two angles that are opposite each other when two lines intersect.
a bcd
In this example, the vertical angles are:
Vertical angles have the same measurement. So:
May 411:04 AM
Using what you know about vertical angles, find the measure of the missing angles.
bc
a
By Vertical Angles: By Supplementary Angles:
May 62:20 PM
7If angle 1 is 60 degrees, what is the measure of angle 3? You must be able to explain why.
21 3
4
Pull
Pull
May 62:20 PM
8If angle 1 is 60 degrees, what is the measure of angle 2? You must be able to explain why.
21
34
Pull
Pull
Adjacent Angles
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.
A
B
C
D
is adjacent to
How do you know?• They have a common side (ray )• They have a common vertex (point B)
What isn't adjacent
Adjacent or Not Adjacent? You Decide!
ab a
b
a
b
Adjacent Not Adjacent Not Adjacentclick to reveal click to reveal click to reveal
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Transversal
Interactive ActivityClick Here
A
PQ
RB
A
E
F
A transversal is a line that cuts across two or more (usually parallel) lines.
Corresponding Angles
Corresponding Angles are on the same side of the transversal and on the same side of the given lines.
ab
c d
e f
g h
In this diagram the corresponding angles are:
Transversal
May 711:06 AM
9Which are pairs of corresponding angles?
A 2 and 6
B 3 and 7
C 1 and 81 2
3 4
5 6
7 8
Pull
Pull
May 711:06 AM
10Which are pairs of corresponding angles?
A 2 and 6
B 3 and 1
C 1 and 8
1
23
4
56
78
Pull
Pull
May 711:58 AM
11Which are pairs of corresponding angles?
A 1 and 5
B 2 and 8
C 4 and 8
1 2
3 4
56
7 8
Pull
Pull
Alternate Exterior Angles
Alternate Exterior Angles are on opposite sides of the transversal and on the outside of the given lines.
ab
c d
e f
g h
In this diagram the alternate exterior angles are:
l
m
n
Which line is the transversal?
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Alternate Interior Angles
Alternate Interior Angles are on opposite sides of the transversal and on the inside of the given lines.
ab
c d
e f
g h
In this diagram the alternate interior angles are: m
n
l
Jun 1710:28 PM
12Which angle corresponds to angle 5?
A B C D
1 3
5 7
2 46 8
m
n
l
Pull
Pull
Jun 1710:28 PM
13What type of angles are and ?
A Alternate Interior Angles B Alternate Exterior Angles C Corresponding Angles D Vertical Angles
1 3
5 7
2 46 8
m
n
l
E Same Side Interior
Pull
Pull B
Jun 1710:28 PM
14What type of angles are and ?
A Alternate Interior Angles B Alternate Exterior Angles C Corresponding Angles D Vertical Angles
1 3
5 7
2 46 8
m
n
l
E Same Side Interior
Pull
Pull E
Jun 1710:28 PM
15What type of angles are and ?
A Alternate Interior Angles B Alternate Exterior Angles C Corresponding Angles D Vertical Angles
1 3
5 7
2 46 8
m
n
l
E Same Side Interior
Pull
Pull C
Jun 212:58 PM
16Are angles 5 and 2 alternate interior angles?
Yes
No Pull
Pull
1 3
5 7
2 46 8
m
n
l
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Jun 212:58 PM
17Are angles 5 and 7 alternate interior angles?
Yes
No
Pull
Pull
1 3
5 7
2 46 8
m
n
l
Jun 212:58 PM
18Are angles 7 and 2 alternate interior angles?
Yes
No
Pull
Pull
1 3
5 7
2 46 8
m
n
l
Jun 212:58 PM
19Are angles 3 and 6 alternate exterior angles?
Yes
No Pull
Pull
1 3
5 7
2 46 8
m
n
l
Jun 21:34 PM
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
SO:1 3
5 7
2 46 8
l
m
n
Jun 1710:36 PM
20Given the measure of one angle, find the measures of as many angles as possible.
4 56
2 71 8
l
m
n
Pull
Pull Angles 5, 7 and 1
Jun 1710:36 PM
21Given the measure of one angle, find the measures of as many angles as possible.
Pull
Pull Angle 4
1 3
5 7
2 48
m
n
l
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perimeter and area of polygons
Perimeter & Circumference
Return to Table of Contents
May 61:14 PM
22What is the Perimeter (P) of the figure?
8 in
Pull
Pull
May 61:14 PM
23What is the Perimeter (P) of the figure?
10 cm
12 cm
8 cm
3 cm
Pull
Pull
May 410:36 AM
24 What is the Circumference (C) of a circle with a radius (r) of 7cm? (Use 3.14 for π)
7 cm
Pull
Pull
May 410:44 AM
25What is the Circumference (C) of a circle with a Diameter (D) of 11in.? (Use 3.14 for π)
11 in.
Pull
Pull
May 410:44 AM
26Find the circumference of a circle whose radius is 2.5 meters. (Use 3.14 for π)
Pull
Pull
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April 23, 2014
May 410:44 AM
27The circumference of a circle is 37.68 cm. What is its radius? (Use 3.14 for π)
Pull
Pull
perimeter and area of polygons
Area of Rectangles
Return to Table of Contents
May 61:08 PM
28What is the Area (A) of the figure?
Pull
Pull
15 ft
6 ft
May 61:15 PM
29Find the area of the figure below.
7 Pull
Pull
May 1010:09 AM
30Dr. 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 AreaB Perimeter
Pull
Pull
May 1012:04 PM
31Now 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? P
ull
Pull
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perimeter and area of polygons
Area of Parallelograms
Return to Table of Contents
May 291:22 PM
Example.
Find the area of the figure.
4 cm
4 cm
2.2 cm 2.2 cm1.9 cm
click to reveal
May 1012:18 PM
32Find the area.
11 ft 10 ft
12 ft
Pull
Pull
Nov 104:15 PM
33Find the area.
17 in
17 in
10 in 12 in
Pull
Pull
12 in
Nov 105:56 PM
34Find the area.
7 m
13 m 13 m
7 m
11 m
Pull
Pull
Nov 105:56 PM
35Find the area.
12 cm
11 cm
9 cm
Pull
Pull
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perimeter and area of polygons
Area of a Triangle
Let's use the same process as we did for the rectangle & parallelogram. How many 1 ft2 tiles fit across the bottom of the triangle?
Area of a Triangle
If we continue to build the triangle with rows of 10 ft2, what happens?
How tall is the triangle? How can you tell?
10 ft
Nov 98:09 AM
How does this help us find the area of the triangle?
Find the area of the rectangle, then divide by 220 ft2
See that the rectangle we built is twice as large as the triangle. How do you find the area of a triangle?
10 ft
4 ft
Nov 98:09 AM
Is this true for all triangles?Let's see!
Calculating base(height) results in 2 triangles!
May 291:21 PM
The Area (A) of a triangle is found by using the formula:
Note: The base & height always form a right angle!
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April 23, 2014
May 291:22 PM
Example.
Find the area of the figure.
4 cm
10 cm 10 cm
6 cm
4
24
12
click to reveal
May 291:22 PM
Try These.
Find the area of the figures.
13 ft
11 ft
9 ft 12 ft 1420
16
15
click to reveal
click to reveal
Nov 104:07 PM
36Find the area.
8 in
5 in
11 in 10 in
Pull
Pull
Nov 104:06 PM
37Find the area
15 m
8 m9 m 12 m
Pull
Pull
perimeter and area of polygons
Area of Trapezoids
Return to Table of Contents
May 64:31 PM
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
See the diagrams below
Base1
Base2
Height
Base1Base2
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April 23, 2014
May 291:21 PM
The Area (A) of a trapezoid is found by using the formula:
Note: The base & height always form a right angle!
May 291:22 PM
Example.
Find the area of the figure.
12 cm
10 cm 11 cm
9 cm
click to reveal
May 291:22 PM
Try These.
Find the area of the figures.
13 ft
11 ft
9 ft 11 ft
20
15
click to reveal
11 ft
click to reveal
9 117
May 1012:26 PM
38Find the area of the trapezoid.
4 m
10 m
6.5 m
Pull
Pull
May 1012:26 PM
39Find the area of the trapezoid.
22 cm
14 cm
8 cm
Pull
Pull
perimeter and area of polygons
Area of Circles
Return to Table of Contents
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May 43:36 PM
Area of a Circle
The Area (A) of a Circle is found by solving the following formula:
May 1011:33 AM
7 cm
Find the area of the circle.A = π r2
1. Substitute the radius into formula.A = π (7)2
2. Use 3.14 as an approximation for π.A = 3.14(49)A = 153.86 cm2
3. Don't forget to label the units as square units.
May 43:59 PM
40What is the Area (A) of a Circle with a radius (r) of 8 m?
8 m
Pull
Pull
May 43:59 PM
41What is the Area (A) of the circle?
Pull
Pull
May 43:59 PM
42A circular sprinkler sprays water with a radius of 11 ft. How much area can the sprinkler cover?
Pull
Pull
May 43:59 PM
43What is the radius of a circle whose area is 254.34 mm2?
Pull
Pull
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May 43:59 PM
44A circular pool has an area of 153.86 ft2.What is its diameter?
Pull
Pull
May 292:51 PM
Mixed Review:Perimeter,
Circumference & Area
Return to Table of Contents
Jun 211:50 PM
45 Find the perimeter and area of the figure.
Pull
Pull5 cm
4 cm 3 cm 4 cm
11 cm
Jun 211:50 PM
46Find the area of the figure.
Pull
Pull
4 yd
8 yd
9 yd
8 yd
Jun 211:50 PM
47 Find the circumference and area of the figure.
Pull
Pull
12 in
Jan 283:16 PM
48If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, how far from the edge of the door should you put the edge of the bar?
Pull
Pull
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Jan 283:16 PM
49A 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?
Pull
Pull
reference sheet
perimeter and area of polygons
Area ofIrregular Figures
Return to Table of Contents
area of irregular figures
Area of Irregular FiguresMethod #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
area of irregular figures
Example:Find the area of the figure.
10 m
6 m
3 m2 m
10 m
6 m
3 m2 m #1
#2
area of irregular figures
Area of Irregular FiguresMethod #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
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April 23, 2014
area of irregular figures
Example:Find the area of the figure.
10 m
6 m
3 m2 m
10 m
6 m
3 m2 m Whole
RectangleExtra
Rectangle
area of irregular figures
Pull
Pull
Pull
Pull
Try These:Find the area of each figure.
2m
4m
2m5m
20 ft
16 ft
8 ft
10 ft
May 1010:37 AM
50Find the area.
4'
2.5'
1.5'
2.5'
8.75'
7.75'
5.25'
Top Rectangle
Bottom Rectangle
Vertical Rectangle
Total Area
May 1010:42 AM
51Find the area.
4 ft.
9 ft.
5 ft.
6 ft.
Side Rectangle
Bottom Right Rectangle
Total Area
Half Circle
perimeter and area of polygons
Area ofShaded Regions
Return to Table of Contents
Dec 28:31 AM
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 units2
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April 23, 2014
Jun 33:09 PM
Example
Find the area of the shaded region.
15 ft
20 ft
7 ft7 ft
Area Whole Rectangle
Area Unshaded Square
Area Shaded Region
Jun 33:09 PM
Try This
Find the area of the shaded region.Area Whole Square
Area Circle
Area Shaded Region14 cm
May 1010:40 AM
52Find the area of the shaded region.
Area Circle
Area Triangle
Area Shaded Region
4 yd
May 1010:40 AM
53A 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
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