Name: ________________________

Geometry Chapter 1: Tools of Geometry

Points, Lines and Planes

Objectives: SWBAT identify Points, Lines, Rays, and Planes.

SWBAT identify Coplanar and Non-Coplanar Points. A. Vocabulary~

1. Point

The basic unit of Geometry.

It is a location

2. Line

An infinite series of points extending in two opposite directions.

Any two points make a line

3. Plane

A flat surface that extends in all directions without end.

Any three points not the same line make a plane

A Written as

A

m

A

B

P Q

RS

Written as

plane PQRS plane QRSP

plane RSPQ plane SPQR

B D

C

k

Written as

plane DBC plane DCB

plane CDB plane CBD

plane BDC plane BCD

plane K

or

Written as

AB BA or line m

n

A

C

E

D

F

H

B

4. Collinear

Points that lie on the same line.

5. Non-Collinear

Points that do not lie on the same line. Must include at least three points

6. Coplanar

Points, lines, segments, or rays that lie on the same plane.

Examples 1. What is another name for ?

Any of the following

2. Give two other names for plane n.

As long as they are any 3 non-collinear points

3. Explain why ABC is not proper way to

name plane n.

They are on the same line, a plane is defined as three points NOT on the same line or

three non-collinear points

4. Tell whether or not the sets are collinear. If so, why or why not.

a. B and F b. BD and E

Yes, any two points make a line Yes, A line extends out forever, and

includes point E as it keeps going

c. EB and A d. plane n and F.

No, A does not lie on the line so No, a plane is made up of 3 non-collinear it cannot be collinear points so by definition it false

, , , ,BD DE ED EB AB

, , ,plane ADB plane BAE plane CBD ect

BD

Refer to the following figure. 5. How many planes are in the figure?

Exactly 5

6. Name three planes.

Plane ABC, Plane ABF, Plane CDFE, Plane DLF, Plane ECA, ect.

7. Name three collinear points.

A, H, and B or B, L, and F

8. Are the points A, H, L and D coplanar? Explain.

No, only A, H, and L lie on the same plane, D is on another plane

No plane can be made using all four of those points.

9. Are the points B, D, and F coplanar? Explain.

Yes, all three points lie on the same plane. Three non-collinear points make a

plane.

10. Nathan’s Mother wants him to go to the post office and the

supermarket. She tells him that the post office, the supermarket, and

their home are collinear. If the post office is between the supermarket and their home; make a map showing the three locations based on this

information.

A

B

C

D

E

F

H L

Linear Measure

Objectives: SWBAT measure segments

SWBAT calculate with measures

A. Vocabulary~

1. Line Segment Part of a line with a definite beginning and end.

2. Line Segment Measure

The measurement or length of a line segment

If you are talking about a segment’s length then NO HAT. If you are

talking about a segment in general terms it HAS A HAT.

3. Congruence

Having the exact same measurement

4. Congruent Segments Segments that have the exact same measurement

Also can be shown by using tick marks

A B

or

Written by

AB BA

0

1

2

M

N

Written by

MN or NM

Written with

A B

C D

Written with

AB CD

A B

C D

5. Segment Addition Postulate

A segment can be accurately measured by the sum of its parts. Add up all the

pieces to make a whole Whole segment = Parts added together

Notice that AB and BC are NOT always congruent, and should NOT be assumed to be congruent unless specified.

Examples: Examples:

1. If AB = 3 cm, and BC = 11 cm, find AC.

2. If CE = 1.1in and ED = 2.7 in, find CD.

3. Find YZ.

A B C

AC

AB BC Whole Sum of the

AC

P rts

BC

a

AB

A B C

C

ED

X

Y

Z

32 mm

15 mm

3 11

14

AC AB BC

AC

AC cm

1.1 2.7

3.8

CD CE ED

CD

CD in

32 15

17

XZ XY YZ

YZ

mm YZ

4. If MO = 32, find the value of each of the following.

a. x =______

Find x first, then we will be able to find the segment measurements. Then we plug “x” back into each given segment

b. MN = _________

c. NO = _________

5. Suppose J is between H and K. Find the length of each segment.

HJ= 2x + 4 JK= 3x + 3

KH= 22

Draw a picture 1st, that will be very helpful. Label your picture second.

Third, Write your equation and solve for x Lastly, Plug x back into the given information

M ONx3 16x

32 3 16

32 4 16

16 4

4

MO MN NO

x x

x

x

x

3

3(4)

12

MN x

MN

MN

16

(4) 16

20

NO x

NO

NO

H J K

22 2 4 3 3

22 5 7

15 5

3

HK HJ JK

x x

x

x

x

2 4

2(3) 4

10

HJ x

HJ

HJ

3 3

3(3) 3

12

JK x

JK

JK

H J K

2x+4 3x+3

22

6. Jason is trying to figure out if he has enough gas to pick up his friends and go to see, “Fast and Furious 18.” Mason has to drive 5.3 miles to

Hank’s House, then 3.4 miles to Stan’s House, and finally 7.2 miles to the movie theatre. If Mason has enough gas to travel 16 miles, will they make

it to the movie theatre or run out of gas on the way?

1. Segment Bisector

A segment, plane or line that cuts a segment in half, or intersects a segment at its midpoint.

2. In the figure, B bisects AC , find the value of c.

Midpoint implies congruence or equal measure. Set it equal to each other and solve.

tan

5.3 3.4 7.2

15.8

,

Journey Hank S Theatre

Journey

Journey

Yes Mason has Enough gas

E

F

GH

A B C

3 4c 5c

3 4 5

4 2

2

c c

c

c

8. In the figure, EF is a bisector of HG , find x and HG.

Bisector bisects or cuts in half. Set the two segments equal to each other and solve.

2 5x

7x

7 2 5

12 7 2 12 5

38

HJ JG HG

x x HG

HG

HG

7 2 5

12

x x

x

Angle Measure

Objectives: SWBAT measure and classify angles.

SWBAT Identify and use congruent angles and angle bisectors.

A. Vocabulary~

1. Ray

Part of a line with a definite beginning, but no end.

Must start notation with beginning of a ray

2. Opposite rays Two rays with a common endpoint

extending out infinitely in opposite directions Has the same properties of a line.

3. Angle

An angle is formed by two non-collinear rays that have a common endpoint.

Sides

The rays of an angle

Vertex

The common endpoint of an angle

Interior of an Angle

Space inside the rays of an angle

Exterior of an Angle

Space outside the rays of an angle

4. Degrees

The unit of measurement for an angle

A B

Written by

AB

A B C

P

Q

R

Side

Side

Vertex 3

124°

3

Written by

QPR

RPQ

P

5. Why are ∠4 and ∠𝑈 not necessarily the same angle?

Type of Angles.

5. RIGHT ANGLE

An angle that measures 90 degrees Right angles will also get a special 90 degree box

6. ACUTE ANGLE

An angle that measures between 0 and 90 degrees

7. OBTUSE ANGLE

An angle that measures between 90 and 180 degrees

8. STRAIGHT ANGLE

An angle that measures 180 degrees

U

&XW XU

XYU

1

U could be a couple different angles

Examples

1. Classify the following angles using a protractor and find the number of degrees.

PJM KJL NJK PJK

Right Angle Acute Angle Obtuse Angle Straight Angle

Everybody look at the clock (wait for kids to find the clock….. this might take a while). Bobby was bored, and so he decided to see what kind of angles are formed by

the two arms of a clock. He looked 6:00 PM, 9:00 PM, 7:10 PM, and 4:30 PM. However, because he was day dreaming and not paying attention, he could not

remember what the differences between a straight, right, acute, and obtuse angles (Karma). Please help out Bobby so he doesn’t look like a bum.

6:00 PM 9:00 PM 7:10 PM 4:30 PM Straight Angle Right Angle Obtuse Angle Acute Angle

PN

M

L

K

J

90 71 141 180

Angle Relationships

Objectives: SWBAT identify and use special pairs of angles.

SWBAT identify perpendicular lines.

1. Vocabulary~

1. Congruent Angles

Angles that have the same measure, or same number of degrees.

These congruent angles can

be labeled using arcs

Congruence

Equal Measure

When do I write it with a ≅ Talking about an angle in general

When do I write it as = When you include an angle measure

Or using a number

Are the following angles congruent? If they are state why, write it in both notations. 1. 2. 3.

55°

55°W

X

Y

Z

Written by

WXY YXZ

Written by

m WXY m YXZ

3 4

No

m m

Yes

ABC DEF

m ABC m DEF

Yes

W Y

m W m Y

Angle Bisector

A ray that cuts an angle into two congruent angles

4. In the Figure, BDbisects CBE . Find “x” and CBD .

Since BDbisects CBE , it cuts the two angles in two equal pieces.

So CBD DBE , and we can set them equal to each other.

Since the angles are congruent, then they

have the same measurement, so CBD is also 56 degrees

5. In the Figure, LK bisects JLM . Find JLK .

Since LK bisects JLM , it cuts the two angles in two equal pieces.

So JLK KLM , and we can set them equal to each other.

56 4

14

CBD DBE

x

x

H

JK

L

M

N

4 15x 6 5x

4 15 6 5

15 2 5

20 2

10

JLK KLM

x x

x

x

x

H

JK

L

M

N

A

B

C

D

E

A

B

C

D

E

56

4x

Vertical Angles

Two nonadjacent angles formed by two intersecting lines. The opposite angles formed by two intersecting lines.

Examples: Solve for the following variables.

1. 2. 3.

3. Solve for x and y. Then find the angle measures.

Notice the vertical angles!

x = _________ y = __________

mAED = _________ mAEC = _________

mCEB = _________ mDEB = _________

12

341 & 3

4 & 2

are opposite angles

are opposite angles

13 10 16 20

10 3 20

30 3

10

y y

y

y

y

3 5 15

2 5 15

2 10

5

x x

x

x

x

6 10

6 5 10

30 10

40

m AEC m DEB

x

13 10

13 10 10

130 10

140

m AED m CEB

y

6 10( )x

13 10y

2 30( )x

16 20y

A

E

C B

D

130x

5 25

5 25

5 5

5

x

x

x

100 4

4 4

96

x

x

Angle Addition Postulate

Objectives: SWBAT apply the AAP to solve for missing angles & variables.

Angle Addition Postulate:

Examples

1. If 𝒎 < 𝑬𝑭𝑯 = 𝟑𝟓 f and 𝒎 < 𝑯𝑭𝑮 = 𝟒𝟎, find the 𝒎 < 𝑬𝑭𝑮.

2. If 𝒎 < 𝟏 = 𝟐𝟐 and 𝒎 < 𝑿𝒀𝒁 = 𝟖𝟔, find the 𝒎 < 𝟐.

3. QR is the angle bisector of PQS. Find all the angle measures not given.

A

B

C

D

35 40

75

m EFH m HFG m EFG

m EFG

m EFG

1 2

22 2 86

22 22

2 64

m m m XYZ

m

m

41 41

82

m PQR m RQS m PQS

m PQS

m PQS

m ABC m CAD m BAD

4. Given that 90m ABC , find is m CBD and m CBE .

5. Given the following, find x and m CAB

2

6 9

81

m CAD x

m CAB x

m DAB

40 4 40 90

4 80 90

4 10

5 2.5

2

m ABE m EBD m CBD m ABC

x

x

x

x or

40

ABE CBD

m CBD

4 40

4(2.5) 40

10 40

50

m EBD m CBD m CBE

x m CBE

m CBE

m CBE

m CBE

2

2

6 9 81

6 72 0

12 6 0

12, 6

6

m CAB m CAD m BAD

x x

x x

x x

x x

x

6 9

6 6 9

45

m CAB

x

m CAB

Angle Relationships

Objectives: SWBAT identify and use special pairs of angles.

SWBAT identify perpendicular lines.

Vocabulary~

Adjacent Angles

Angles that share a common ray. They are side by side (Apartment Angles)

Linear Pair

Adjacent angles that form a straight angle

Complementary Angles

Angles that add up to 90 degrees.

Complementary Adjacent Complementary Nonadjacent

Complementary Angles can be adjacent or don’t have to be adjacent.

Supplementary Angle

Angles that add up to 180 degrees.

They don’t have to be adjacent. So a linear pair will Always be supplementary but unless they are adjacent supplementary will not be a linear Pair.

Supplementary Adjacent Supplementary Nonadjacent

A linear pair and supplementary angles will ALWAYS make a straight line. That is a

Good way to spot a straight angle / linear pair/ supplementary angles.

A

B

C

D

R

S

T

U

1 2

1

2

30° 60°

135°

45°

Examples:

1. Use the diagram on the right.

a. Are 3 and 5 adjacent angles? No because they don’t share a side.

b. Are 1 and 2 adjacent angles? Yes, because they share a ray.

c. Are 1 and 2 a linear pair? No, they don’t make a straight angle.

d. Are 3 and 4 a linear pair? Yes, they are adjacent, and form a straight angle.

e. Are 2 and 4 vertical angles? No, the ray divides the angle opposite to <4

f. Are 3 and 5 vertical angles? Yes

g. If m3 = 45 then m4=________. <3 & <4 are a linear pair

h. If m5 = 53 then m3=________. <3 and <5 are vertical angles

2. Use the diagram to the right.

a. Name two pairs of complementary angles

b. What kind of angles are <RWS and <TWS?

c. What angle is supplementary <TWU?

d. Are <RWV and <VWU a linear pair? Explain why or why not.

No because they are not supplementary

3 4 180

45 4 180

135 4

m m

m

m

5

1 2

3

4

3 5

53 5

m m

m

RWQ& QWV , VEU & UWT

supplementary

UWS

3. Given that A and B are complementary with mA = 3x + 5 and mB = 7x + 15. Solve for x and find the measures of A and B. 4. Given that E is supp. to F . If mE = 15x + 16 and mF = 4x + 12, solve for x

and find the measures of E and F

5. Given the diagram to the right, answer the following questions. a) Can angle DBE have a complement? Why?

No, It equals 90 degrees so we can’t add another angle and Have it equal 90 (0 degrees is not an angle)

b) Are there any vertical angles present in this diagram?

No, there are no vertical angles because two lines don’t Intersect and continue past the intersection point.

c) Can there be any non-adjacent complementary angles in the diagram?

No, only adjacent angles in this diagram.

d) Even though there are no numbers, why can you assume there is at least one linear pair?

Because you can always assume a line is straight and make 180 degrees

3 5 7 15 90

10 20 90

10 70

7

x x

x

x

x

3 5

3 7 5

21 5

26

m A

x

( )

7 15

7 7 15

49 15

64

m B

x

( )

15 16 4 12 180

19 28 180

19 152

8

x x

x

x

x

15 16

15 8 16

120 16

136

m F

x

( )

4 12

4 8 12

32 12

44

m E

x

( )