Determining if a Triangle is Possible - PBworksasaymath.pbworks.com/w/file/fetch/79256816/2D...

2D_Geometry_Presentationabridged.notebook

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April 23, 2014

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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April 23, 2014

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


3

April 23, 2014

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


4

April 23, 2014

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


5

April 23, 2014

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


A 2 and 6

B 3 and 1

C 1 and 8

1

23

4

56

78

Pull

Pull

May 711:58 AM


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?

http://www.mathopenref.com/transversal.html


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April 23, 2014

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



1 3

5 7

2 46 8

m

n

l


Pull

Pull E

Jun 1710:28 PM



1 3

5 7

2 46 8

m

n

l


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


7

April 23, 2014

Jun 212:58 PM


Yes

No

Pull

Pull

1 3

5 7

2 46 8

m

n

l

Jun 212:58 PM


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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April 23, 2014

perimeter and area of polygons

Perimeter & Circumference


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


9

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


Area of Rectangles


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


10

April 23, 2014


Area of Parallelograms


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


11

April 23, 2014


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!


12

April 23, 2014

May 291:22 PM

Example.


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


Area of Trapezoids


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.


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


Area of Circles



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April 23, 2014

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


15

April 23, 2014

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


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


16

April 23, 2014

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


Area ofIrregular Figures


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


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 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


Example:Find the area of the figure.

10 m

6 m

3 m2 m

10 m

6 m

3 m2 m Whole

RectangleExtra

Rectangle


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


Area ofShaded Regions


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


18

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

Determining if a Triangle is Possible - PBworksasaymath.pbworks.com/w/file/fetch/79256816/2D...

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