Goal Your Notes VOCABULARY - rjssolutions.com · Your Notes UNDEFINED TERMS Point A point has no dimension. A It is represented by a dot . point A Line A line has one dimension. A

Copyright © Holt McDougal. All rights reserved. Lesson 1.1 • Geometry Notetaking Guide 1

Identify Points, Lines, and Planes

1.1

Your Notes

Goal p Name and sketch geometric figures.

VOCABULARY

Undefined term

Point

Line

Plane

Collinear points

Coplanar points

Defined Terms

Line segment, endpoints

Ray

Opposite rays

Intersection


Identify Points, Lines, and Planes

1.1

Your Notes

Goal p Name and sketch geometric figures.

VOCABULARY

Undefined term A word without a formal definition

Point A point has no dimension. It is represented by a dot.

Line A line has one dimension. It is represented by a line with two arrowheads.

Plane A plane has two dimensions. It is represented by a shape that looks like a floor or a wall.

Collinear points Points that lie on the same line

Coplanar points Points that lie in the same plane

Defined Terms Terms that can be described using known words

Line segment, endpoints Part of a line that consists of two points, called endpoints, and all the points on the line between the endpoints

Ray The ray AB consists of the endpoint A and all

points on @##$ AB that lie on the same side of A as B.

Opposite rays If point C lies on @##$ AB between A and

B, then ###$ CA and ###$ CB are opposite rays.

Intersection The intersection of two or more geometric figures is the set of points that the figures have in common.

LAH_GE_11_FL_NTG_Ch01_001-030.indd 1LAH_GE_11_FL_NTG_Ch01_001-030.indd 1 1/24/09 3:38:32 PM1/24/09 3:38:32 PM

Your NotesUNDEFINED TERMS

Point A point has dimension. A

point AIt is represented by a .

Line A line has dimension. A

B

line l, line AB(AB),

or line BA(BA)

It is represented by a with two arrowheads, but it extends without end.

Through any points, there isexactly line. You can use any

points on a line to name it.

Plane A plane has dimensions. M

A

plane M or plane ABC

BCIt is represented by a shape that

looks like a floor or wall, but it extends without end.

Through any points not on the same line, there is exactly plane. You can use points that are not all on the same line to name a plane.

a. Give two other names for @##$ LN .

Lb

a

MN

P

Give another name for plane Z.

b. Name three points that are collinear. Name four points that are coplanar.

a. Other names for @##$ LN are and . Other names for plane Z are plane and .

b. Points lie on the same line, so they are collinear. Points lie on the same plane, so they are coplanar.

Example 1 Name points, lines, and planes

1. Give two other names for @##$ MQ . Name a point that is not coplanar with points L, N, and P.

Checkpoint Use the diagram in Example 1.

There is a line through points L and Q that is not shown in the diagram. Try to imagine what plane LMQ would look like if it were shown.

2 Lesson 1.1 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Your NotesUNDEFINED TERMS

Point A point has no dimension. A

point AIt is represented by a dot .

Line A line has one dimension. A

B

line l, line AB(AB),

or line BA(BA)

It is represented by a line with two arrowheads, but it extends without end.

Through any two points, there isexactly one line. You can use any two points on a line to name it.

Plane A plane has two dimensions. M

A

plane M or plane ABC

BCIt is represented by a shape that

looks like a floor or wall, but it extends without end.

Through any three points not on the same line, there is exactly one plane. You can use three points that are not all on the same line to name a plane.

a. Give two other names for @##$ LN .

Lb

a

MN

P

Give another name for plane Z.

b. Name three points that are collinear. Name four points that are coplanar.

a. Other names for @##$ LN are @###$ LM and line b . Other names for plane Z are plane LMP and LNP .

b. Points L, M, and N lie on the same line, so they are collinear. Points L, M, N, and P lie on the same plane, so they are coplanar.

Example 1 Name points, lines, and planes

1. Give two other names for @##$ MQ . Name a point that is not coplanar with points L, N, and P.

@###$ QM and line a; point Q


There is a line through points L and Q that is not shown in the diagram. Try to imagine what plane LMQ would look like if it were shown.




Your NotesDEFINED TERMS: SEGMENTS AND RAYS

Line AB (written as ) and points B

line

AA and B are used here to definethe terms below.

Segment The line segment AB, or segmentAB, (written as ) consists of the

B

segment

endpoint endpoint

A

endpoints A and B and all points on @##$ AB that are A and B.

Note that } AB can also be named .

Ray The ray AB (written as )

B

ray

endpoint

A

B

endpoint

A

consists of the endpoint A and allpoints on @##$ AB that lie on the sameside of as .

Note that ###$ AB and ###$ BA are rays.

a. Give another name for } VX.

b. Name all rays with endpoints W. YX

W

ZVWhich of these rays are opposite rays?

a. Another name for } VX is .

b. The rays with endpoint W are .

The opposite rays with endpoint W are ,

and .

Example 2 Name segments, rays, and opposite rays

2. Give another name for } YW .

3. Are ###$ VX and ###$ XV the same ray? Are ###$ VW and ###$ VX the same ray? Explain.


In Example 2, ###$ WY and ###$ WX have a common

, but are not

. So they are not opposite rays.


Your NotesDEFINED TERMS: SEGMENTS AND RAYS

Line AB (written as @##$ AB ) and points B

line

AA and B are used here to definethe terms below.

Segment The line segment AB, or segmentAB, (written as } AB ) consists of the

B

segment

endpoint endpoint

A

endpoints A and B and all points on @##$ AB that are between A and B.

Note that } AB can also be named } BA .

Ray The ray AB (written as ###$ AB )

B

ray

endpoint

A

B

endpoint

A

consists of the endpoint A and allpoints on @##$ AB that lie on the sameside of A as B .

Note that ###$ AB and ###$ BA are different rays.

a. Give another name for } VX.

b. Name all rays with endpoints W. YX

W

ZVWhich of these rays are opposite rays?

a. Another name for } VX is } XV .

b. The rays with endpoint W are ####$ WV , ####$ WY , ####$ WX , and ####$ WZ .

The opposite rays with endpoint W are ####$ WV and ####$ WX ,

and ####$ WY and ####$ WZ .

Example 2 Name segments, rays, and opposite rays

2. Give another name for } YW .

} WY

3. Are ###$ VX and ###$ XV the same ray? Are ###$ VW and ###$ VX the same ray? Explain.

No, the rays do not have the same endpoint; Yes, the rays have a common endpoint, are collinear, and consist of the same points.


In Example 2, ###$ WY and ###$ WX have a common endpoint , but are not collinear . So they are not opposite rays.


Your Notes

a. Sketch a plane and a line that intersects the plane at more than one point.

b. Sketch a plane and a line that is in the plane. Sketch another line that intersects the line and plane at a point.

a. b.

Example 3 Sketch intersections of lines and planes

Sketch two planes that intersect in a line.

Step 1 Sketch one plane as if you are facing it.

Step 2 Sketch a second plane that is . Use dashed lines to show where one plane is hidden.

Step 3 Sketch the line of .

Example 4 Sketch intersections of planes

4. Sketch two different lines that intersect a plane at different points.

5. Name the intersection of @##$ MX

D

C

aM

X

Yand line a.

6. Name the intersection ofplane C and plane D.

Checkpoint Complete the following exercises.

Homework

Sketching a diagram is a way to show geometric relationships informally. A sketch can be made by hand without using measuring tools.


Your Notes

a. Sketch a plane and a line that intersects the plane at more than one point.

b. Sketch a plane and a line that is in the plane. Sketch another line that intersects the line and plane at a point.

a. M b. M

pq

Example 3 Sketch intersections of lines and planes

Sketch two planes that intersect in a line.

Step 1 Sketch one plane as if you 1

2

3 mare facing it.

Step 2 Sketch a second plane that is horizontal . Use dashed lines to show where one plane is hidden.

Step 3 Sketch the line of intersection .

Example 4 Sketch intersections of planes

4. Sketch two different lines M

p

qthat intersect a plane at different points.

5. Name the intersection of @##$ MX

D

C

aM

X

Yand line a.

point M

6. Name the intersection ofplane C and plane D.

line a


Homework

Sketching a diagram is a way to show geometric relationships informally. A sketch can be made by hand without using measuring tools.




Your Notes

1.2 Use Segments and CongruenceGoal p Use segment postulates to identify congruent

segments.

VOCABULARY

Postulate, axiom

Theorem

Coordinate

Distance

Between

Congruent segments

POSTULATE 1 RULER POSTULATE

The points on a line can be B

names of points

coordinates of points

A

x2x1

matched one to one with real numbers. The real number that corresponds to a point is the

of the point.

The between points B

AB 5 ⏐x2 2 x1⏐

A

x2x1A and B, written as AB, is the absolute value of the difference of the coordinates of A and B.


Your Notes

1.2 Use Segments and CongruenceGoal p Use segment postulates to identify congruent

segments.

VOCABULARY

Postulate, axiom A rule that is accepted without proof

Theorem A rule that can be proved

Coordinate The real number that corresponds to a point

Distance The distance between two points A and B, written as AB, is the absolute value of the difference of the coordinates of A and B.

Between When three points are collinear, you can say that one point is between the other two.

Congruent segments Line segments that have the same length

POSTULATE 1 RULER POSTULATE

The points on a line can be B

names of points

coordinates of points

A

x2x1

matched one to one with real numbers. The real number that corresponds to a point is the coordinate of the point.

The distance between points B

AB 5 ⏐x2 2 x1⏐

A

x2x1A and B, written as AB, is the absolute value of the difference of the coordinates of A and B.


Your Notes


Measure the length of } CD to the DC

nearest tenth of a centimeter.

SolutionAlign one mark of a metric ruler with C. Then estimate the coordinate of D. For example, if you align C with 1, D appears to align with .

DC

cm 1 2 3 4 5 6 7 8

CD 5⏐ 2 ⏐5 Ruler postulate

The length of } CD is about centimeters.

Example 1 Apply the Ruler Postulate

POSTULATE 2 SEGMENT ADDITION POSTULATE

If B is between A and C, then

B CA

AC

AB BC

AB 1 BC 5 AC.

If AB 1 BC 5 AC, then B is between A and C.

Road Trip The locations Starting point

R

D

S

Rest area64 mi

87 miDestination

shown lie in a straight line. Find the distance from the starting point to the destination.

SolutionThe rest area lies between the starting point and the destination, so you can apply the Segment Addition Postulate.

SD 5 1 Segment Addition Postulate

5 1 Substitute for and .

5 Add.

The distance from the starting point to the destination is miles.

Example 2 Apply the Segment Addition Postulate

A ruler is a drawing tool that allows you to measure segment lengths. You can also use a ruler to draw segments of given lengths.

Your Notes


Measure the length of } CD to the DC

nearest tenth of a centimeter.

SolutionAlign one mark of a metric ruler with C. Then estimate the coordinate of D. For example, if you align C with 1, D appears to align with 4.7 .

DC

cm 1 2 3 4 5 6 7 8

CD 5⏐ 4.7 2 1 ⏐5 3.7 Ruler postulate

The length of } CD is about 3.7 centimeters.

Example 1 Apply the Ruler Postulate

POSTULATE 2 SEGMENT ADDITION POSTULATE

If B is between A and C, then

B CA

AC

AB BC

AB 1 BC 5 AC.

If AB 1 BC 5 AC, then B is between A and C.

Road Trip The locations Starting point

R

D

S

Rest area64 mi

87 miDestination

shown lie in a straight line. Find the distance from the starting point to the destination.

SolutionThe rest area lies between the starting point and the destination, so you can apply the Segment Addition Postulate.

SD 5 SR 1 RD Segment Addition Postulate

5 64 1 87 Substitute for SR and RD .

5 151 Add.

The distance from the starting point to the destination is 151 miles.

Example 2 Apply the Segment Addition Postulate

A ruler is a drawing tool that allows you to measure segment lengths. You can also use a ruler to draw segments of given lengths.


Your Notes

Homework

Use the diagram to find KL. K LJ

38

15

Use the Segment Addition Postulate to write an equation. Then solve the equation to find KL.

5 1 KL Segment Addition Postulate

5 1 KL Substitute for and .

5 KL Subtract from each side.

Example 3 Find a length

Plot F(4, 5), G(21, 5), H(3, 3), and J(3, 22) in a coordinate plane. Then determine whether } FG and } HJ are congruent.

Horizontal segment: Subtract the

x

y

1

1

of the endpoints.

FG 5⏐ ⏐5

Vertical segment: Subtract the of the endpoints.

HJ 5⏐ ⏐5

} FG and } HJ have the length. So } FG } HJ .

Example 4 Compare segments for congruence

1. Find the length of } AB to the nearest 1 } 8 inch. Then

draw a segment with the same length.

A B

2. Find QS and PQ. Q R SP

55

31 30

3. Consider the points A(22, 21), B(4, 21), C(3, 0), and D(3, 5). Are } AB and } CD congruent?



Your Notes

Homework

Use the diagram to find KL. K LJ

38

15

Use the Segment Addition Postulate to write an equation. Then solve the equation to find KL.

JL 5 JK 1 KL Segment Addition Postulate

38 5 15 1 KL Substitute for JL and JK .

23 5 KL Subtract 15 from each side.

Example 3 Find a length

Plot F(4, 5), G(21, 5), H(3, 3), and J(3, 22) in a coordinate plane. Then determine whether } FG and } HJ are congruent.

Horizontal segment: Subtract the

x

y

1

1

G(21, 5) F(4, 5)

H(3, 3)

J(3, 22)

x-coordinates of the endpoints.

FG 5⏐ 4 2 (21) ⏐5 5

Vertical segment: Subtract the y-coordinates of the endpoints.

HJ 5⏐ 3 2 (22) ⏐5 5

} FG and } HJ have the same length. So } FG > } HJ .

Example 4 Compare segments for congruence

1. Find the length of } AB to the nearest 1 } 8 inch. Then

draw a segment with the same length.

A B

1 7 } 8 inches; Check drawing

2. Find QS and PQ. Q R SP

55

31 30 61; 24

3. Consider the points A(22, 21), B(4, 21), C(3, 0), and D(3, 5). Are } AB and } CD congruent? No




Your Notes

1.3


Use Midpoint and Distance FormulasGoal p Find lengths of segments in the coordinate plane.

VOCABULARY

Midpoint

Segment bisector

Find RS. SR T

V

W

21.7

Solution

Point is the midpoint of } RS . So, RT 5 5 21.7.

RS 5 1 Segment Addition Postulate

5 1 Substitute.

5 Add.

The length of } RS is .

Example 1 Find segments lengths

1. Find AB. D

A C B

2 14

Checkpoint Complete the following exercise.

Your Notes

1.3


Use Midpoint and Distance FormulasGoal p Find lengths of segments in the coordinate plane.

VOCABULARY

Midpoint The point that divides a segment into two congruent segments

Segment bisector A point, ray, line, line segment, or plane that intersects the segment at its midpoint

Find RS. SR T

V

W

21.7

Solution

Point T is the midpoint of } RS . So, RT 5 TS 5 21.7.

RS 5 RT 1 TS Segment Addition Postulate

5 21.7 1 21.7 Substitute.

5 43.4 Add.

The length of } RS is 43.4 .

Example 1 Find segments lengths

1. Find AB. D

A C B

2 14

4 1 } 2



Your Notes

Point C is the midpoint of } BD . B C D

3x 2 2 2x 1 3

Find the length of } BC .

SolutionStep 1 Write and solve an equation.

BC 5 CD Write equation.

5 Substitute.

5 Subtract from each side.

x 5 Add to each side.

Step 2 Evaluate the expression for BC when x 5 .

BC 5 5 5

So, the length of } BC is units.

Example 2 Use algebra with segment lengths

2. Point K is the midpoint of } JL . Find the length of } KL .

J K L

8 2 3x 2x 1 5


THE MIDPOINT FORMULA

The coordinates of the

x

y

x1 x2

y2

y1 1 y

2

2

y1

y1 1 y

2

2

x1 1 x

2

2

x1 1 x

2

2

B(x2, y2)

A(x1, y1)( ),

midpoint of a segment are the averages of thex-coordinates and of they-coordinates of the endpoints.

If A(x1, y1)and B(x2, y2) are points in a coordinate plane, then the midpoint M of } AB has coordinates

1 x1 1 x2 } 2 , y1 1 y2 } 2 2 .


Your Notes

Point C is the midpoint of } BD . B C D

3x 2 2 2x 1 3

Find the length of } BC .

SolutionStep 1 Write and solve an equation.

BC 5 CD Write equation.

3x 2 2 5 2x 1 3 Substitute.

x 2 2 5 3 Subtract 2x from each side.

x 5 5 Add 2 to each side.

Step 2 Evaluate the expression for BC when x 5 5 .

BC 5 3x 2 2 5 3(5) 2 2 5 13

So, the length of } BC is 13 units.

Example 2 Use algebra with segment lengths

2. Point K is the midpoint of } JL . Find the length of } KL .

J K L

8 2 3x 2x 1 5

6 1 } 5


THE MIDPOINT FORMULA

The coordinates of the

x

y

x1 x2

y2

y1 1 y

2

2

y1

y1 1 y

2

2

x1 1 x

2

2

x1 1 x

2

2

B(x2, y2)

A(x1, y1)( ),

midpoint of a segment are the averages of thex-coordinates and of they-coordinates of the endpoints.

If A(x1, y1)and B(x2, y2) are points in a coordinate plane, then the midpoint M of } AB has coordinates

1 x1 1 x2 } 2 , y1 1 y2 } 2 2 .



Your Notes


a. Find Midpoint The endpoints of } PR are P(22, 5) and R(4, 3). Find the coordinates of the midpoint M.

b. Find Endpoint The midpoint of } AC is M(3, 4). One endpoint is A(1, 6). Find the coordinates of endpoint C.

Solutiona. Use the Midpoint Formula.

x

y

1

1

P(22, 5)

R(4, 3)

M(?, ?)

M1 1 ,

1 2 5 M( , )

The coordinates of the midpoint of } PR are .

b. Let (x, y) be the coordinates of

x

y

1

1

A(1, 6)

M (3, 4)

C (x, y)

endpoint C. Use the Midpoint Formula to find x and y.

Step 1 Find x. Step 2 Find y.

1 x

2 5

1 y

2 5

1 x 5 1 y 5

x 5 y 5

The coordinates of endpoint C are .

Example 3 Use the Midpoint Formula

Multiply each side of the equation by the denominator to clear the fraction.

3. The endpoints of } CD are C(28, 21) and D(2, 4). Find the coordinates of the midpoint M.

4. The midpoint of } XZ is M(5, 26). One endpoint is X(23, 7). Find the coordinates of endpoint Z.


Your Notes


a. Find Midpoint The endpoints of } PR are P(22, 5) and R(4, 3). Find the coordinates of the midpoint M.

b. Find Endpoint The midpoint of } AC is M(3, 4). One endpoint is A(1, 6). Find the coordinates of endpoint C.

Solutiona. Use the Midpoint Formula.

x

y

1

1

P(22, 5)

R(4, 3)

M(?, ?)

M1 22 1 4

2,

5 1 3

2 2 5 M( 1 , 4 )

The coordinates of the midpoint of } PR are M(1, 4) .

b. Let (x, y) be the coordinates of

x

y

1

1

A(1, 6)

M (3, 4)

C (x, y)

endpoint C. Use the Midpoint Formula to find x and y.

Step 1 Find x. Step 2 Find y.

1 1 x

2 5 3

6 1 y

2 5 4

1 1 x 5 6 6 1 y 5 8

x 5 5 y 5 2

The coordinates of endpoint C are (5, 2) .

Example 3 Use the Midpoint Formula

Multiply each side of the equation by the denominator to clear the fraction.

3. The endpoints of } CD are C(28, 21) and D(2, 4). Find the coordinates of the midpoint M.

M 1 23, 3 } 2 2

4. The midpoint of } XZ is M(5, 26). One endpoint is X(23, 7). Find the coordinates of endpoint Z.

(13, 219)



Your Notes


THE DISTANCE FORMULA

If A(x1, y1) and B(x2, y2)

x

yB(x2, y2)

A(x1, y1)

C (x2, y1)

⏐y2 2 y1⏐

⏐x2 2 x1⏐

are points in a coordinate plane, then the distance between A and B is

AB 5 Ï}}}

( )2 1 ( )2

What is the approximate length

x

y

1

1

T (24, 3)

R(3, 2)

of } RT , with endpoints R(3, 2) and T(24, 3)?

SolutionUse the Distance Formula.

RT 5 Ï}}}

( )2 1 ( )2 Distance Formula

5 Ï}}}

( 2 3)2 1 ( 2 )2 Substitute.

5 Ï}}

( )2 1 ( )2 Subtract.

5 Ï}}

1 Evaluate powers.

5 Ï}

Add.

ø Use a calculator.

The length of } RT is about .

Example 4 Use the Distance Formula

The symbol ø means "is approximately equal to."

5. What is the approximate length of } GH , with endpoints G(5, 21) and H(23, 6)?


Homework

Your Notes


THE DISTANCE FORMULA

If A(x1, y1) and B(x2, y2)

x

yB(x2, y2)

A(x1, y1)

C (x2, y1)

⏐y2 2 y1⏐

⏐x2 2 x1⏐

are points in a coordinate plane, then the distance between A and B is

AB 5 Ï}}}

( x2 2 x1 )2 1 ( y2 2 y1 )2

What is the approximate length

x

y

1

1

T (24, 3)

R(3, 2)

of } RT , with endpoints R(3, 2) and T(24, 3)?

SolutionUse the Distance Formula.

RT 5 Ï}}}

( x2 2 x1 )2 1 ( y2 2 y1 )2 Distance Formula

5 Ï}}}

( 24 2 3)2 1 ( 3 2 2 )2 Substitute.

5 Ï}}

( 27 )2 1 ( 1 )2 Subtract.

5 Ï}}

49 1 1 Evaluate powers.

5 Ï}

50 Add.

ø 7.07 Use a calculator.

The length of } RT is about 7.07 .

Example 4 Use the Distance Formula

The symbol ø means "is approximately equal to."

5. What is the approximate length of } GH , with endpoints G(5, 21) and H(23, 6)?

about 10.63


Homework


Your Notes


1.4 Measure and Classify AnglesGoal p Name, measure, and classify angles.

VOCABULARY

Angle

Sides of an angle

Vertex of an angle

Measure of an angle

Acute angle

Right angle

Obtuse angle

Straight angle

Congruent angles

Angle bisector

Name the three angles in the diagram.

, or C

A

D

B, or

, or

Example 1 Name anglesYou should not name any of these angles B because all three angles have B as their

.

Your Notes


1.4 Measure and Classify AnglesGoal p Name, measure, and classify angles.

VOCABULARY

Angle An angle consists of two different rays with the same endpoint.

Sides of an angle In an angle, the rays are called the sides of the angle.

Vertex of an angle In an angle, the endpoint is the vertex of the angle.

Measure of an angle In ∠AOB, ####$ OA and ###$ OB can be matched one to one with real numbers from 0 to 180. The measure of ∠AOB is equal to the absolute value of the difference between the real numbers for ####$ OA and ###$ OB .

Acute angle An angle that measures between 08 and 908

Right angle An angle that measures 908

Obtuse angle An angle that measures between 908 and 1808

Straight angle An angle that measures 1808

Congruent angles Angles with the same measure

Angle bisector A ray that divides an angle into two angles that are congruent

Name the three angles in the diagram.

∠ABC , or ∠CBA C

A

D

B ∠CBD , or ∠DBC

∠ABD , or ∠DBA

Example 1 Name anglesYou should not name any of these angles B because all three angles have B as their vertex .


Your Notes


POSTULATE 3: PROTRACTOR POSTULATE

Consider ###$ OB and point A on 9090

8010070

11060120

50130

4014

0

3015

0

2016

0

10 170

0 180

10080

11070 12060 13050 14040 15030

1602017010

1800

A

BO

one side of ###$ OB . The rays of the form ###$ OA can be matched one to one with the real numbers from 0 to .

The measure of is equal to between the real numbers

for ###$ OA and ###$ OB .

Use the diagram to find the 9090

8010070

11060120

50130

4014

0

3015

0

2016

0

10 170

0 180

10080

11070 12060 13050 14040 15030

1602017010

1800

WV

TSR

measure of the indicated angle.Then classify the angle.

a. ∠WSR b. ∠TSW

c. ∠RST d. ∠VST

a. ###$ SR is lined up with the 08 on the scale of the protractor. ###$ SW passes through on the

scale. So, m∠WSR 5 . It is angle.

b. ###$ ST is lined up with the 08 on the scale of the protractor. ###$ SW passes through on the

scale. So, m∠TSW 5 . It is angle.

c. m∠RST 5 . It is angle.

d. m∠VST 5 . It is angle.

Example 2 Measure and classify angles

1. Name all the angles in

H

JF

Gthe diagram at the right.

2. What type of angles do the x-axis and y-axis form in a coordinate plane?


A protractor is a drawing tool that allows you to measure angles. You can also use a protractor to drawangles of given measures.

Your Notes


POSTULATE 3: PROTRACTOR POSTULATE

Consider ###$ OB and point A on 9090

8010070

11060120

50130

4014

0

3015

0

2016

0

10 170

0 180

10080

11070 12060 13050 14040 15030

1602017010

1800

A

BO

one side of ###$ OB . The rays of the form ###$ OA can be matched one to one with the real numbers from 0 to 180 .

The measure of ∠AOB is equal to the absolute value of the difference between the real numbers

for ###$ OA and ###$ OB .

Use the diagram to find the 9090

8010070

11060120

50130

4014

0

3015

0

2016

0

10 170

0 180

10080

11070 12060 13050 14040 15030

1602017010

1800

WV

TSR

measure of the indicated angle.Then classify the angle.

a. ∠WSR b. ∠TSW

c. ∠RST d. ∠VST

a. ###$ SR is lined up with the 08 on the outer scale of the protractor. ###$ SW passes through 658 on the outer

scale. So, m∠WSR 5 658 . It is an acute angle.

b. ###$ ST is lined up with the 08 on the inner scale of the protractor. ###$ SW passes through 1158 on the inner

scale. So, m∠TSW 5 1158 . It is an obtuse angle.

c. m∠RST 5 1808 . It is a straight angle.

d. m∠VST 5 908 . It is a right angle.

Example 2 Measure and classify angles

1. Name all the angles in

H

JF

Gthe diagram at the right.

∠FGH or ∠HGF, ∠FGJ or ∠JGF, ∠JGH or ∠HGJ

2. What type of angles do the x-axis and y-axis form in a coordinate plane?

right angles


A protractor is a drawing tool that allows you to measure angles. You can also use a protractor to drawangles of given measures.


Your NotesPOSTULATE 4: ANGLE ADDITION POSTULATE

Words If P is in the interior of

P

R

S

m/RSP

m/PST

m/RST

T

∠RST, then the measure of ∠RST is equal to the sum of the measures of ∠ and ∠ .

Symbols If P is in the interior of ∠RST, then m∠RST 5 m∠ 1 m∠ .

Given that m∠GFJ 5 1558, H

J

FG

(4x 2 1)8

(4x 1 4)8

find m∠GFH and m∠HFJ.

Solution

Step 1 Write and solve an equation to find the value of x.

m∠GFJ 5 m∠ 1 m∠ Angle Addition Postulate

5 ( )8 1 ( )8 Substitute.

5 Combine like terms.

5 Subtract from each side.

5 x Divide each side by .

Step 2 Evaluate the given expressions when x 5 .

m∠GFH 5 ( )8 5 ( )8 5 .

m∠HFJ 5 ( )8 5 ( )8 5 .

So, m∠GFH 5 and m∠HFJ 5 .

Example 3 Find angle measures

A point is in the interior of an angle if it is between points that lie on each side of the angle.

interior

3. Given that ∠VRS is a right angle, V

SR

T

(3x 1 2)8

(x 2 4)8

find m∠VRT and ∠TRS.



Your NotesPOSTULATE 4: ANGLE ADDITION POSTULATE

Words If P is in the interior of

P

R

S

m/RSP

m/PST

m/RST

T

∠RST, then the measure of ∠RST is equal to the sum of the measures of ∠ RSP and ∠ PST .

Symbols If P is in the interior of ∠RST, then m∠RST 5 m∠ RSP 1 m∠ PST .

Given that m∠GFJ 5 1558, H

J

FG

(4x 2 1)8

(4x 1 4)8

find m∠GFH and m∠HFJ.

Solution

Step 1 Write and solve an equation to find the value of x.

m∠GFJ 5 m∠ GFH 1 m∠ HFJ Angle Addition Postulate

1558 5 ( 4x 1 4 )8 1 ( 4x 2 1 )8 Substitute.

155 5 8x 1 3 Combine like terms.

152 5 8x Subtract 3 from each side.

19 5 x Divide each side by 8 .

Step 2 Evaluate the given expressions when x 5 19 .

m∠GFH 5 ( 4x 1 4 )8 5 ( 4 p 19 1 4 )8 5 808 .

m∠HFJ 5 ( 4x 2 1 )8 5 ( 4 p 19 2 1 )8 5 758 .

So, m∠GFH 5 808 and m∠HFJ 5 758 .


A point is in the interior of an angle if it is between points that lie on each side of the angle.

interior

3. Given that ∠VRS is a right angle, V

SR

T

(3x 1 2)8

(x 2 4)8

find m∠VRT and ∠TRS.

m∠VRT 5 198, m∠TRS 5 718




Your Notes


Identify all pairs of congruent angles

P N

ML

in the diagram. If m∠P 5 1208, whatis m∠N?

SolutionThere are two pairs of congruent angles:

∠P > and ∠L >

Because ∠P > , m∠P 5 . So, m∠N 5 .

Example 4 Identify congruent angles

Homework

In the diagram at the right, ###$ WY

WX

YZ

bisects ∠XWZ, and m∠XWY 5 298.Find m∠XWZ.

SolutionBy the Angle Addition Postulate,m∠XWZ 5 1 .

Because ###$ WY bisects ∠XWZ, you know > .

So, 5 , and you can write m∠XWZ 5 1 5 1 5 .

Example 5 Double an angle measure

4. Identify all pairs of congruent

A D

CB

angles in the diagram. If m∠B 5 1358, what is m∠D?

5. In the diagram below, ###$ KM bisects

KN

M

L

∠LKN and m∠LKM 5 788. Find m∠LKN.


Your Notes


Identify all pairs of congruent angles

P N

ML

in the diagram. If m∠P 5 1208, whatis m∠N?

SolutionThere are two pairs of congruent angles:

∠P > ∠N and ∠L > ∠M

Because ∠P > ∠N , m∠P 5 m∠N . So, m∠N 5 1208 .

Example 4 Identify congruent angles

Homework

In the diagram at the right, ###$ WY

WX

YZ

bisects ∠XWZ, and m∠XWY 5 298.Find m∠XWZ.

SolutionBy the Angle Addition Postulate,m∠XWZ 5 m∠XWY 1 m∠YWZ .

Because ###$ WY bisects ∠XWZ, you know ∠XWY > ∠YWZ .

So, m∠XWY 5 m∠YWZ , and you can write m∠XWZ 5 m∠XWY 1 m∠YWZ 5 298 1 298 5 588 .

Example 5 Double an angle measure

4. Identify all pairs of congruent

A D

CB

angles in the diagram. If m∠B 5 1358, what is m∠D?

∠B > ∠D and ∠A > ∠C; 1358

5. In the diagram below, ###$ KM bisects

KN

M

L

∠LKN and m∠LKM 5 788. Find m∠LKN.

1568



Your Notes

1.5


Describe Angle Pair RelationshipsGoal p Use special angle relationships to find angle

measures.

VOCABULARY

Complementary angles

Supplementary angles

Adjacent angles

Linear pair

Vertical angles

In the figure, name a pair of

E

D

B

A

C

388

1288528

complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

Solution

Because 1 5 908, and are angles.

Because 1 5 1808, and are angles.

Because and share a common vertex and side, they are angles.

Example 1 Identify complements and supplements

In Example 1, ∠BDE and ∠CDE share a common vertex. But they share common

points, so they are not adjacent angles.

Your Notes

1.5


Describe Angle Pair RelationshipsGoal p Use special angle relationships to find angle

measures.

VOCABULARY

Complementary angles Two angles whose sum is 908

Supplementary angles Two angles whose sum is 1808

Adjacent angles Two angles that share a common vertex or side, but have no common interior points

Linear pair Two adjacent angles are a linear pair if their noncommon sides are opposite rays.

Vertical angles Two angles are vertical angles if their sides form two pairs of opposite rays.

In the figure, name a pair of

E

D

B

A

C

388

1288528


Solution

Because 528 1 388 5 908, ∠ABD and ∠CDB are complementary angles.

Because 528 1 1288 5 1808, ∠ABD and ∠EDB are supplementary angles.

Because ∠CDB and ∠BDE share a common vertex and side, they are adjacent angles.

Example 1 Identify complements and supplements

In Example 1, ∠BDE and ∠CDE share a common vertex. But they share common interior points, so they are not adjacent angles.


Your Notes


Example 2 Find measures of complements and supplements

a. Given that ∠1 is a complement of ∠2 and m∠2 5 578, find m∠1.

b. Given that ∠3 is a supplement of ∠4 and m∠4 5 418, find m∠3.

Solutiona. You can draw a diagram with

5782

1

complementary adjacent angles to illustrate the relationship.

m∠1 5 2 5 2 5

b. You can draw a diagram with

41843supplementary adjacent angles to illustrate the relationship.

m∠3 5 2 5 2 5

Angles are sometimes named with numbers. An angle measure in a diagram has a degree symbol. An angle name does not.

1. In the figure, name a pair of

1408408

508

DF

G

E

A

CB


2. Given that ∠1 is a complement of ∠2 and m∠1 5 738, find m∠2.

3. Given that ∠3 is a supplement of ∠4 andm∠4 5 378, find m∠3.


Your Notes


Example 2 Find measures of complements and supplements

a. Given that ∠1 is a complement of ∠2 and m∠2 5 578, find m∠1.

b. Given that ∠3 is a supplement of ∠4 and m∠4 5 418, find m∠3.

Solutiona. You can draw a diagram with

5782

1

complementary adjacent angles to illustrate the relationship.

m∠1 5 908 2 m∠2 5 908 2 578 5 338

b. You can draw a diagram with

41843supplementary adjacent angles to illustrate the relationship.

m∠3 5 1808 2 m∠4 5 1808 2 418 5 1398

Angles are sometimes named with numbers. An angle measure in a diagram has a degree symbol. An angle name does not.

1. In the figure, name a pair of

1408408

508

DF

G

E

A

CB


complementary: ∠DEF and ∠ABC; supplementary; ∠FEG and ∠ABC;adjacent: ∠DEF and ∠FEG

2. Given that ∠1 is a complement of ∠2 and m∠1 5 738, find m∠2.

178

3. Given that ∠3 is a supplement of ∠4 andm∠4 5 378, find m∠3.

1438



Your Notes


Basketball The basketball pole

B C

A

(3x 1 8)8 (4x 2 3)8

D

forms a pair of supplementary angles with the ground. Find m∠BCA and m∠DCA.

Solution

Step 1 Use the fact that is the sum of the measuresof supplementary angles.

m∠BCA 1 m∠DCA 5 Write equation.

( )8 1 ( )8 5 Substitute.


5 Subtract.

5 Divide.

Step 2 Evaluate the original expressions when x 5 .

m∠BCA 5 ( )8 5 ( )8 5 .

m∠DCA 5 ( )8 5 ( )8 5 .

The angle measures are and .


In a diagram, you can assume that a line that looks straight is straight. In Example 3, B, C, and D lie on @##$ BD . So, ∠BCD is a

angle.

4. In Example 3, suppose the angle measures are (5x 1 1)8 and (6x 1 3)8. Find m∠BCA and m∠DCA.


Your Notes


Basketball The basketball pole

B C

A

(3x 1 8)8 (4x 2 3)8

D

forms a pair of supplementary angles with the ground. Find m∠BCA and m∠DCA.

Solution

Step 1 Use the fact that 1808 is the sum of the measuresof supplementary angles.

m∠BCA 1 m∠DCA 5 1808 Write equation.

( 3x 1 8 )8 1 ( 4x 2 3 )8 5 1808 Substitute.

7x 1 5 5 180 Combine like terms.

7x 5 175 Subtract.

x 5 25 Divide.

Step 2 Evaluate the original expressions when x 5 25 .

m∠BCA 5 ( 3x 1 8 )8 5 ( 3 p 25 1 8 )8 5 838 .

m∠DCA 5 ( 4x 2 3 )8 5 ( 4 p 25 2 3 )8 5 978 .

The angle measures are 838 and 978 .


In a diagram, you can assume that a line that looks straight is straight. In Example 3, B, C, and D lie on @##$ BD . So, ∠BCD is a straight angle.

4. In Example 3, suppose the angle measures are (5x 1 1)8 and (6x 1 3)8. Find m∠BCA and m∠DCA.

818 and 998



Your Notes


Identify all of the linear pairs 12

453

and all of the vertical angles in the figure at the right.

SolutionTo find vertical angles, look for angles formed by .

and are vertical angles.

To find linear pairs, look for adjacent angles whose noncommon sides are .

and are a linear pair. and are a linear pair.

Example 4 Identify angle pairs

In the diagram, one side of ∠1 and one side of ∠4 are opposite rays. But the angles are not a linear pair because they are not

.

5. Identify all of the linear pairs 12

4 5 63and all of the vertical angles

in the figure.


Two angles form a linear pair. The measure of one angle is 4 times the measure of the other. Find the measure of each angle.

x 8SolutionLet x8 be the measure of one angle. The measure of the other angle is . Then use the fact that the angles of a linear pair are to write an equation.

1 5 Write an equation.


5 Divide each side by .

The measures of the angles are and 5 .

Example 5 Find angle measures in a linear pair

One angle measure should be four times the other angle measure. The measures should also have a sum of 180° for the angles to form a linear pair.

Your Notes


Identify all of the linear pairs 12

453

and all of the vertical angles in the figure at the right.

SolutionTo find vertical angles, look for angles formed by intersecting lines .

∠1 and ∠3 are vertical angles.

To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays .

∠1 and ∠2 are a linear pair. ∠2 and ∠3 are a linear pair.

Example 4 Identify angle pairs

In the diagram, one side of ∠1 and one side of ∠4 are opposite rays. But the angles are not a linear pair because they are not adjacent .

5. Identify all of the linear pairs 12

4 5 63and all of the vertical angles

in the figure.

linear pairs: none; vertical angles: ∠1 and ∠4, ∠2 and ∠5, ∠3 and ∠6


Two angles form a linear pair. The measure of one angle is 4 times the measure of the other. Find the measure of each angle.

x 8SolutionLet x8 be the measure of one angle. The measure of the other angle is 4x8 . Then use the fact that the angles of a linear pair are supplementary to write an equation.

x8 1 4x8 5 1808 Write an equation.

5x 5 180 Combine like terms.

x 5 36 Divide each side by 5 .

The measures of the angles are 368 and 4(368) 5 1448 .

Example 5 Find angle measures in a linear pair

4x8

One angle measure should be four times the other angle measure. The measures should also have a sum of 180° for the angles to form a linear pair.


Your Notes


Homework

CONCEPT SUMMARY: INTERPRETING A DIAGRAM

There are some things you can

A B

DE

C

conclude from a diagram, and some you cannot. For example, here are some things that you can conclude from the diagramat the right.

• All points shown are .

• Points A, B, and C are , and B is between A and C.

• @##$ AC , ###$ BD , and ###$ BE at point B.

• ∠DBE and ∠EBC are angles, and ∠ABC is a .

• Point E lies in the of ∠DBC.

In the diagram above, you cannot

A B

DE

C

conclude that } AB > } BC , that ∠DBE > ∠EBC, or that ∠ABD is a right angle. This information must be indicated, as shown at the right.

6. Two angles form a linear pair. The measure of one angle is 3 times the measure of the other. Find the meaure of each angle.


Your Notes


Homework

CONCEPT SUMMARY: INTERPRETING A DIAGRAM

There are some things you can

A B

DE

C

conclude from a diagram, and some you cannot. For example, here are some things that you can conclude from the diagramat the right.

• All points shown are coplanar .

• Points A, B, and C are collinear , and B is between A and C.

• @##$ AC , ###$ BD , and ###$ BE intersect at point B.

• ∠DBE and ∠EBC are adjacent angles, and ∠ABC is a straight angle .

• Point E lies in the interior of ∠DBC.

In the diagram above, you cannot

A B

DE

C

conclude that } AB > } BC , that ∠DBE > ∠EBC, or that ∠ABD is a right angle. This information must be indicated, as shown at the right.

6. Two angles form a linear pair. The measure of one angle is 3 times the measure of the other. Find the meaure of each angle.

458 and 1358



Your Notes

1.6 Classify PolygonsGoal p Classify polygons.

VOCABULARY

Polygon

Sides

Vertex

Convex

Concave

n-gon

Equilateral

Equiangular

Regular


Your Notes

1.6 Classify PolygonsGoal p Classify polygons.

VOCABULARY

Polygon A polygon is a closed plane figure with the following properties: (1) It is formed by three or more line segments called sides. (2) Each side intersects exactly two sides, one at each endpoint, so that no two sides with a common endpoint are collinear.

Sides The sides of a polygon are the line segments that form the polygon.

Vertex A vertex of a polygon is an endpoint of a side of the polygon.

Convex A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon.

Concave A concave polygon is a polygon that is not convex.

n-gon An n-gon is a polygon with n sides.

Equilateral A polygon is equilateral if all of its sides are congruent.

Equiangular A polygon is equiangular if all of its angles in the interior are congruent.

Regular A polygon is regular if all sides and all angles are congruent.



Your NotesIDENTIFYING POLYGONS

In geometry, a figure that lies in a plane is called a plane figure. A is a closed plane figure with the following properties.

1. It is formed by three or more line segments called .

2. Each side intersects exactly sides, one at each endpoint, so that no two sides with a common endpoint are .

Each endpoint of a side is a of the C

D

EA

Bpolygon. The plural of vertex is vertices. A polygon can be named by listing the vertices in consecutive order. For example, ABCDE and CDEAB are both correct names for the polygon at the right.

Tell whether the figure is a polygon and whether it is convex or concave.

a. b. c.

Solutiona. Some segments intersect more than two segments, so

it is .

b. The figure is .

c. The figure is .

Example 1 Identify polygons

1. 2.

Checkpoint Tell whether the figure is a polygon and whether it is convex or concave.

A plane figure is two-dimensional. Later, you will study three-dimensional space figures such as prisms and cylinders.


Your NotesIDENTIFYING POLYGONS

In geometry, a figure that lies in a plane is called a plane figure. A polygon is a closed plane figure with the following properties.

1. It is formed by three or more line segments called sides .

2. Each side intersects exactly two sides, one at each endpoint, so that no two sides with a common endpoint are collinear .

Each endpoint of a side is a vertex of the C

D

EA

Bpolygon. The plural of vertex is vertices. A polygon can be named by listing the vertices in consecutive order. For example, ABCDE and CDEAB are both correct names for the polygon at the right.

Tell whether the figure is a polygon and whether it is convex or concave.

a. b. c.

Solutiona. Some segments intersect more than two segments, so

it is not a polygon .

b. The figure is a convex polygon .

c. The figure is a concave polygon .

Example 1 Identify polygons

1. 2.

convex polygon not a polygon

Checkpoint Tell whether the figure is a polygon and whether it is convex or concave.

A plane figure is two-dimensional. Later, you will study three-dimensional space figures such as prisms and cylinders.



Your Notes


Homework

Classify the polygon by the number of sides. Tell which terms apply to the polygon: equilateral, equiangular, regular, or not regular.

Solution

The polygon has sides. It is equilateral and equiangular, so it is a .

Example 2 Classify polygons

The head of a bolt is shaped like (4x 1 3) mm

(5x 2 1) mm

a regular hexagon. The expressions shown represent side lengths of the hexagonal bolt. Find the length of a side.

SolutionFirst, write and solve an equation to find the value of x. Use the fact that the sides of a regular hexagon are .

5 Write an equation.

5 Simplify.

Then evaluate one of the expressions to find a side length when x 5 . 4x 1 3 5 4( ) 1 3 5

The length of a side is millimeters.

Example 3 Find side lengths

Check that this is a reasonable side length for a hexagonal bolt.

3. Classify the polygon by the number of sides. Tell which terms apply to the polygon: equilateral, equiangular, regular, or not regular.

4. The expressions (4x 1 8)8 and (5x 2 5)8 represent the measures of two of the congruent angles in Example 3. Find the measure of an angle.


Your Notes


Homework

Classify the polygon by the number of sides. Tell which terms apply to the polygon: equilateral, equiangular, regular, or not regular.

Solution

The polygon has 8 sides. It is equilateral and equiangular, so it is a regular octagon .

Example 2 Classify polygons

The head of a bolt is shaped like (4x 1 3) mm

(5x 2 1) mm

a regular hexagon. The expressions shown represent side lengths of the hexagonal bolt. Find the length of a side.

SolutionFirst, write and solve an equation to find the value of x. Use the fact that the sides of a regular hexagon are congruent .

4x 1 3 5 5x 2 1 Write an equation.

4 5 x Simplify.

Then evaluate one of the expressions to find a side length when x 5 4 . 4x 1 3 5 4( 4 ) 1 3 5 19

The length of a side is 19 millimeters.

Example 3 Find side lengths

Check that this is a reasonable side length for a hexagonal bolt.

3. Classify the polygon by the number of sides. Tell which terms apply to the polygon: equilateral, equiangular, regular, or not regular.

quadrilateral; not regular

4. The expressions (4x 1 8)8 and (5x 2 5)8 represent the measures of two of the congruent angles in Example 3. Find the measure of an angle.

608



Your Notes

1.7 Find Perimeter, Circumference, and AreaGoal p Find dimensions of polygons.

Tennis The in-bounds portion of a singles tennis

78 ft

27 ft

court is shown. Find its perimeter and area.

Perimeter Area

P 5 2l 1 2w A 5 lw

5 2( ) 1 2( ) 5 ( )

5 5

The perimeter is ft and the area is ft2.

Example 1 Find the perimeter and area of a rectangle

1. In Example 1, the width of the in-bounds rectangle increases to 36 feet for doubles play. Find the perimeter and area of the in-bounds rectangle.



FORMULAS FOR PERIMETER P, AREA A, AND CIRCUMFERENCE C

Square

s

Rectangle

w

side length s length l and width w

P 5 P 5 A 5 A 5

Triangle

a

b

ch

Circle rside lengths a, b, radius r

and c, base b, C 5 and height h. A 5

P 5 Pi (π) is the ratio of a

A 5 circle's circumference to its diameter.

Your Notes

1.7 Find Perimeter, Circumference, and AreaGoal p Find dimensions of polygons.

Tennis The in-bounds portion of a singles tennis

78 ft

27 ft

court is shown. Find its perimeter and area.

Perimeter Area

P 5 2l 1 2w A 5 lw

5 2( 78 ) 1 2( 27 ) 5 78 ( 27 )

5 210 5 2106

The perimeter is 210 ft and the area is 2106 ft2.

Example 1 Find the perimeter and area of a rectangle

1. In Example 1, the width of the in-bounds rectangle increases to 36 feet for doubles play. Find the perimeter and area of the in-bounds rectangle.

perimeter: 228 ft, area: 2808 ft2



FORMULAS FOR PERIMETER P, AREA A, AND CIRCUMFERENCE C

Square

s

Rectangle

w

side length s length l and width w

P 5 4s P 5 2l 1 2w A 5 s2 A 5 lw

Triangle

a

b

ch

Circle rside lengths a, b, radius r

and c, base b, C 5 2πr and height h. A 5 πr2 P 5 a 1 b 1 c Pi (π) is the ratio of a

A 5 1 } 2 bh

circle's circumference to its diameter.


Your Notes


Archery The smallest circle on an Olympic target is 12 centimeters in diameter. Find the approximate circumference and area of the smallest circle.

SolutionFirst find the radius. The diameter is 12 centimeters, so the radius is 1 } 2 ( ) 5 centimeters.

Then find the circumference and area. Use 3.14 for π.

P 5 2πr ø 2( )( ) 5

A 5 πr2 ø ( )2 5

Example 2 Find the circumference and area of a circle

2.

8 m

Checkpoint Find the approximate circumference and area of the circle.

The approximations3.14 and 22 }

7 are

commonly used as approximations for the irrational number π. Unless told otherwise, use 3.14 for π.

Triangle JKL has vertices J(1, 6), K(6, 6), and L(3, 2). Find the approximate perimeter of triangle JKL.

SolutionFirst draw triangle JKL in a coordinate

x

y

1

1

plane. Then find the side lengths. Because } JK is horizontal, use the

to find JK. Use the to find JL and LK.

JK 5⏐ 2 ⏐5 units

JL 5 Ï}}}

( 2 1)2 1 (2 2 )2 5 Ï}

ø units

LK 5 Ï}}}

( 2 3)2 1 ( 2 2)2 5 Ï}

5 units

Then find the perimeter.

P 5 JK 1 JL 1 LK ø 1 1 5 units.

Example 3 Using a coordinate plane

Write down your calculations to make sure you do not make a mistake substituting values in the Distance Formula.

Your Notes


Archery The smallest circle on an Olympic target is 12 centimeters in diameter. Find the approximate circumference and area of the smallest circle.

SolutionFirst find the radius. The diameter is 12 centimeters, so the radius is 1 } 2 ( 12 ) 5 6 centimeters.

Then find the circumference and area. Use 3.14 for π.

P 5 2πr ø 2( 3.14 )( 6 ) 5 37.68 cm

A 5 πr2 ø 3.14 ( 6 )2 5 113.04 cm2

Example 2 Find the circumference and area of a circle

2.

8 m

C ø 50.24 m; A ø 200.96 m2

Checkpoint Find the approximate circumference and area of the circle.

The approximations3.14 and 22 }

7 are

commonly used as approximations for the irrational number π. Unless told otherwise, use 3.14 for π.

Triangle JKL has vertices J(1, 6), K(6, 6), and L(3, 2). Find the approximate perimeter of triangle JKL.

SolutionFirst draw triangle JKL in a coordinate

x

y

1

1

J(1,6) K(6, 6)

L(3, 2)

plane. Then find the side lengths. Because } JK is horizontal, use the Ruler Postulate to find JK. Use the Distance Formula to find JL and LK.

JK 5⏐ 6 2 1 ⏐5 5 units

JL 5 Ï}}}

( 3 2 1)2 1 (2 2 6 )2 5 Ï}

20 ø 4.5 units

LK 5 Ï}}}

( 6 2 3)2 1 ( 6 2 2)2 5 Ï}

25 5 5 units

Then find the perimeter.

P 5 JK 1 JL 1 LK ø 5 1 4.5 1 5 5 14.5 units.

Example 3 Using a coordinate plane

Write down your calculations to make sure you do not make a mistake substituting values in the Distance Formula.


Your Notes

Lawn care You are using a roller to smooth a lawn. You can roll about 125 square yards in one minute. About how many minutes does it take to roll a lawn that is 120 feet long and 75 feet wide?

SolutionYou can roll the lawn at a rate of 125 square yards per minute. So, the amount of time it takes you to roll the lawn depends on its .

Step 1 Find the area of the rectangular lawn.

Area 5 lw 5 ( ) 5 ft2

The rolling rate is in square yards per minute. Rewrite the area of the lawn in square yards. There are feet in 1 yard, and 2 5 square feet in one square yard.

9000 ft2 p 1 yd2

ft2 5 yd2 Use unit analysis.

Step 2 Write a verbal model to represent the situation. Then write and solve an equation based on the verbal model.

Let t represent the total time (in minutes) needed to roll the lawn.

Area of lawn

(yd2) 5 Rolling rate

(yd2 per min) 3 Total time

(min)

5 p t Substitute.

5 t Divide each side by .

It takes about minutes to roll the lawn.

Example 4 Solve a multi-step problem


You can roll about 1250 yards in 10 minutes. Use this fact to check that your solution is reasonable for the lawn area you found in Step 1.

Your Notes

Lawn care You are using a roller to smooth a lawn. You can roll about 125 square yards in one minute. About how many minutes does it take to roll a lawn that is 120 feet long and 75 feet wide?

SolutionYou can roll the lawn at a rate of 125 square yards per minute. So, the amount of time it takes you to roll the lawn depends on its area .

Step 1 Find the area of the rectangular lawn.

Area 5 lw 5 120 ( 75 ) 5 9000 ft2

The rolling rate is in square yards per minute. Rewrite the area of the lawn in square yards. There are 3 feet in 1 yard, and 3 2 5 9 square feet in one square yard.

9000 ft2 p 1 yd2

9 ft2 5 1000 yd2 Use unit analysis.

Step 2 Write a verbal model to represent the situation. Then write and solve an equation based on the verbal model.

Let t represent the total time (in minutes) needed to roll the lawn.

Area of lawn

(yd2) 5 Rolling rate

(yd2 per min) 3 Total time

(min)

1000 5 125 p t Substitute.

8 5 t Divide each side by 125 .

It takes about 8 minutes to roll the lawn.

Example 4 Solve a multi-step problem


You can roll about 1250 yards in 10 minutes. Use this fact to check that your solution is reasonable for the lawn area you found in Step 1.


Your Notes


The base of a triangle is 24 feet. Its area is 216 square feet. Find the height of the triangle.

h

24 ft

Solution

A 5 Area of a triangle

5 Substitute.

5 Multiply.

5 h Solve for h.

The height is feet.

Example 5 Find unknown length

3. Find the perimeter of the

x

y

1

1A(1, 1)

B(7, 6)

C(7, 3)

triangle shown at the right.

4. Suppose a lawn is half as long and half as wide as the lawn in Example 4. Will it take half the time to roll the lawn? Explain.

5. The area of a triangle is 96 square inches, and its height is 8 inches. Find the length of its base.


Homework

In Example 5, you may want to make and label a sketch of the triangle. The sketch will not be exact, but it will help you visualize the given information.

Your Notes


The base of a triangle is 24 feet. Its area is 216 square feet. Find the height of the triangle.

h

24 ft

Solution

A 5 1 } 2 bh Area of a triangle

216 5 1 } 2 (24)(h) Substitute.

216 5 12h Multiply.

18 5 h Solve for h.

The height is 18 feet.

Example 5 Find unknown length

3. Find the perimeter of the

x

y

1

1A(1, 1)

B(7, 6)

C(7, 3)

triangle shown at the right.

about 17.1 units

4. Suppose a lawn is half as long and half as wide as the lawn in Example 4. Will it take half the time to roll the lawn? Explain.

No, it will take a quarter of the time to roll the lawn because it is a quarter of the original area.

5. The area of a triangle is 96 square inches, and its height is 8 inches. Find the length of its base.

24 inches


Homework

In Example 5, you may want to make and label a sketch of the triangle. The sketch will not be exact, but it will help you visualize the given information.


Words to ReviewGive an example of the vocabulary word.

28 Words to Review • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Point, line, plane

Coplanar points

Ray

Intersection

Coordinate

Collinear points


Opposite rays

Postulate, axiom

Distance

Words to ReviewGive an example of the vocabulary word.


Point, line, plane

point

planeline

Coplanar points

DT

D and T are coplanar points.

Ray

X

Y

###$ XY is a ray with initial point X.

Intersection

P

The intersection of two different lines is a point.

CoordinateBA

x2x1

The coordinates of points A and B are x1 and x2.

Collinear points

A B

A and B are collinear points.


C

D

}

CD is a line segment with endpoints C and D.

Opposite rays

AB

C

If C is between A and B, then ###$ CA and ###$ CB are opposite rays.

Postulate, axiom

A postulate, or axiom, is a rule that is accepted without proof.

DistanceB

AB 5 ⏐x2 2 x1⏐

A

x2x1

The distance between points A and B is

⏐x2 2 x1⏐.


Between

Midpoint

Angle, sides, vertex

Acute angle

Obtuse angle

Congruent segments

Segment bisector

Measure of an angle

Right angle

Straight angle

Copyright © Holt McDougal. All rights reserved. Words to Review • Geometry Notetaking Guide 29

Between

A BC

Point C is between Points A and B.

Midpoint

ABM

M is the midpoint of } AB .

Angle, sides, vertex

C

B

A

Sides ###$ AB and ###$ AC form ∠A. The vertex is A.

Acute angle

A

08 < m∠A < 908

Obtuse angle

A

908 < m∠A < 1808

Congruent segments

A B

C D

} AB and }

CD are congruent.

Segment bisector

A

F

G

B

@##$ FG is a segment bisector of } AB .

Measure of an angle

A408

The measure of ∠A is 408.

Right angle

A

m∠A 5 908

Straight angle

A

m∠A 5 1808

Copyright © Holt McDougal. All rights reserved. Words to Review • Geometry Notetaking Guide 29


Review your notes and Chapter 1 by using the Chapter Review on pages 61–64 of your textbook.

Angle bisector, congruent angles

Complementary angles,adjacent angles

Polygon, side, vertex

n-gon

Supplementary angles, linear pair

Vertical angles

Concave, convex

Equilateral, equiangular, regular


Review your notes and Chapter 1 by using the Chapter Review on pages 61–64 of your textbook.

Angle bisector, congruent angles

B

C D

E

###$ BD is an angle bisector of ∠CBE.

∠CBD and ∠DBE are congruent.

Complementary angles,adjacent angles

P

R

S

∠QPR and ∠RPS are complementary.

∠QPR and ∠RPS are adjacent.

Polygon, side, vertex

side

polygon

vertex

n-gon

An n-gon is a polygon with n sides.

Supplementary angles, linear pair

TS

X

Y

∠STX and ∠XTY are supplementary.

∠STX and ∠XTY are a linear pair.

Vertical angles

21

∠1 and ∠2 are vertical angles.

Concave, convex

ABCD is

F G

HJ

A

B

CDconcave.

FGHJ is convex.

Equilateral, equiangular, regular

The polygon is equilateral and equiangular, so it is regular.



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Goal Your Notes VOCABULARY - rjssolutions.com · Your Notes UNDEFINED TERMS Point A point has no dimension. A It is represented by a dot . point A Line A line has one dimension. A