Chapter 1 Student Notes

Chapter 1 TestTuesday, August 29th

1.1 Points, Lines and Planes

Point -

A

B

C

D

m

Line -

Collinear -

• T / F A and B are Collinear

• T / F A and C are Collinear

• T / F A, B and C are Collinear

A B

C

P

A

B C

Plane -

Coplanar -A B

C D

G

E F

• Name 3 Coplanar Points ________

•Name 3 Noncoplanar Points _________

• T/F C, D and G are coplanar

• T/F A, B, E, F are coplanar

• T/F A, B, C, E are coplanar

Draw and Label each of the following1. n and m intersect at P

2. p contains N

3. P contains A and B, but not C

Draw and Label each of the following4. m intersects P at X

5. P and R intersect at m

1.2Segments

Objective:1) Learn the language of Geometry2) Become familiar with segments and segment

measure

Line Segment -

A

B

Betweenness of Points -

A

B C

Measure of a Segment -

M

N6

Segment Congruence -

R

S7

T

U7

Segment Congruence is marked on a figure in the following

manner.

CB

A

1212

Multiple Pairs of Congruent Segments

From the markings on the above figure, make 2 congruence statement.CB

DA

1. AC = 4, AD = 3, Find CD = ______

2. CD = 15, AD = 7, Find AC = _____

A is between C and D. Find Each Measure.

C 4 A 3 D

C A 7 D

15

3. AC = x + 1, AD = x + 3, CD = 3x – 5, Find x = _____

A is between C and D. Find Each Measure.

C x + 1 A x + 3 D

3x - 5

1. AC = 8, AD = 5, Find CD = ______

2. CD = 20, AD = 12, Find AC = _____

A is between C and D. Find Each Measure.

C 8 A 5 D

C A 12 D

20

3. AC = 2x + 1, AD = 2x + 3, CD = 5x – 10, Find x = ___

A is between C and D. Find Each Measure.

C 2x + 1 A 2x + 3 D

5x – 10

C is between A and B in each figure. Select the figure that has AB = 12. Select all that apply.

A 8 C 4 B D 8 B A

A. C is between A and B. B. B is between A and D.

D B A

C. B is between A and D.AB = 2x + 5, BD = 3x + 4, AD = 6x – 3

D B A

D. B is between A and D.AB = 2x + 2, DB = 4x +2, DA =34

Answer: ____________

1.3Distance and Midpoint

Distance on a Number Line =

A B C D

-5 0 5

AB =

BC =

AD =

BD =

Use the number line to find the length of each segment.

Distance on a Coordinate Plane

A(2, 2)B(-4, 1)

C(2, -4)

AB =

Find the length of each segment.

DistanceFormula

A(2, 2)B(-4, 1)

C(2, -4)

Find the length of each segment.

BC

Midpoint on a Number Line

Midpoint =

A B C D

-5 0 5

1. AB

Find the midpoint of each segment.

2. AD

A B C D

-5 0 5

Find the midpoint of each segment.

3. BC

4. If A is the midpoint of EC, what is the location for point E?

Midpoint on a Coordinate Plane

A(2, 2)B(-4, 1)

C(2, -4)

Midpoint = ( )x1 + x2 , y1 + y2

2 2

Find the midpoint of each segment.

1. AB

= ( ) = ( )

Midpoint on a Coordinate Plane

A(2, 2)B(-4, 1)

C(2, -4)

Find the midpoint of each segment.

1. BC

= ( ) = ( )

Midpoint on a Coordinate Plane

A(2, 2)B(-4, 1)

C(2, -4)

Find the midpoint of each segment.

2. AC

= ( ) = ( )

M is the midpoint of AB. Given the following information, find the missing coordinates.

M(2, 6) , B(12, 10) , A ( ? , ? ) Midpoint = ( )x1 + x2 , y1 + y2

2 2

M is the midpoint of AB. Given the following information, find the missing coordinates.

M(6, -8) , A(2, 0) , B ( ? , ? )Midpoint = ( )x1 + x2 , y1 + y2

2 2

1.4Angle Measure

Ray -

E

DS

R

BA

Angle–

Angles and Points

Points _______________________________

G ____________________

H ____________________

E ____________________H

D

E

G

F

Naming Angles

H

D

E

G

F2

1. ________

2. ________

3. ________

4. ________

Name the angle at the right as many ways as possible.

Naming Angles

32

J

K

M

L

Name the angles at the right as many ways as possible.

1. _______

2. _______

3. _______

4. _______

1. _______

2. _______

3. _______

4. _______

Naming Angles

32

J

K

M

L

Name the angles at the right as many ways as possible.

1. _________

2. _________

3. _________

●●● ●

●●

There is more than one angle at vertex K, __________________ ____________________________________

Types of Angles

Right angle:

________ different types of angles:

Acute angle:

Types of Angles

Obtuse angle: Straight angle:

Can also be called __________ ________________.

Congruent Angles

33o

33oM

W

Multiple Sets of Congruent Angles

__________

__________

A B

CD

KM is an angle bisector.

Angle Bisector

64

J

K

M

L

What conclusion can you draw about the figure at the right?

_________________

or

________________

When you want to add angles, use ______________________ _____________________________________________________________..

If you add m1 + m2, what is your result?_____________________________.

Adding Angles

●●●

21

J

K

M

L28o48o

Angle Addition Postulate The sum of the two smaller angles adjacent angles will

_______________________________________________________________________________________________.

Complete:

m ______ + m ______ = m _______

orm ______ + m ______ = m _______

21

R

S

U

T

Draw your own diagram and answer this question:

If ML is an angle bisector of PMY and mPML = 87, then find:

mPMY = _______mLMY = _______

Example

JK is an angle bisector of LJM. mLJK = 4x + 10, mKJM = 6x – 4. Find x and mLJM.

L

J M

K(4x + 10)o

(6x – 4)o

mLJM = _____

RS is an angle bisector of PRT. mPRT = 11x – 12, mSRT = 4x + 3. Find x and mPRS.

P

R T

S

(4x + 3)o

mPRS = ___

1-5Angle Pairs

Complementary Angles -

Examples:

21

M

N T

DR

S

Perpendicular – _______________________

Supplementary Angles-

Examples:

J

GL

K

21

KH

Adjacent Angles

Adjacent Angles

43

Vertical Angles-

Example:

A

E

D

C

B

12

4

3

Theorem:

●●● A

E

D

C

B

12

4

3

What’s “Important” in Geometry?4 things to always look for !

. . . and ___________( )Most of the rules (theorems)and vocabulary of Geometryare based on these 4 things.

Examples1. 1 & 2 are complementary. m1 = 4x + 5,

m2 = 5x + 4. Find x and the measure of each angle.

x = _____

m1 = _____ m2

= _____

Examples

2. 5 & 6 are supplementary. m5 = 10x + 12,

m6 = 2x + 6. Find x and the measure of each angle.

x = _____

m5 = _____ m6

= _____

2 1 3

4

Examples

3. m1 = 2x + 7, m3 = 3x – 3. Find x and the measure of each angle.

Find x = _____

m 2 = _____

m1 = _____

2 1 3

4

Examples4. m2 = 5x + 12, m4 = 7x – 20. Find x and the measure of each angle.

x = _____

m 2 = _____

m1 = _____

1.6Polygons

Determine if each figure is a polgyon

Polygon -

Example of Concave PolygonsConcave Polgons

Examples of Convex Polygons

Convex Polygons

Number of Sides3456789

10111213n

Name of Polygon Hint

Examples of Regular Polygons

Regular Polygon-

Find the perimeter of each polygon.

Perimeter - distance around a polygon

Square RectangleRegular Hexagon

P = _______

8in

6cm

3cm

P = ______

4ft

P = ________

Name each polygon by its number of sides. Then classify it as concave or convex and regular or irregular.

Name each polygon by its number of sides. Then classify it as concave or convex and regular or irregular.

Find the perimeter and area of the polygon below.

3cm

3cm

8cm

8cm

5cm

5cm

5cm5cm

P = ________

A = ________

1. A(-3, 0), B(0, 4), C(4, -3)

Triangle ABC has the following coordinates. Find the perimeter of ABC.

P = _______