Tools of Geometry - WYSIWYG - homewysiwyg.cmswiki.wikispaces.net/file/view/Chapter 1 - Tools of... · 1 Tools of Geometry ... E m n p C A D B 5. ... Describe the intersection of planes

C H A P T E R

1 Tools of Geometry

1.1 Let's Get This StartedPoints, Lines, Planes, Rays, and

Segments | p. 3

1.2 All About AnglesNaming Angles, Classifying Angles,

Duplicating Angles, and Bisecting

Angles | p. 13

1.3 Special AnglesComplements, Supplements,

Midpoints, Perpendiculars, and

Perpendicular Bisectors | p. 27

1.4 A Little Dash of LogicTwo Methods of Logical

Reasoning | p. 39

1.5 ConditionalsConditional Statements, Postulates,

and Theorems | p. 45

1.6 Forms of ProofParagraph Proof, Two-Column Proof,

Construction Proof, and Flow Chart

Proof | p. 55

A protractor is a basic tool used since ancient times to measure angles. A

compass is a drawing instrument used to create arcs and circles. The bottom

of a semicircular protractor can serve as a straightedge. Using a protractor,

compass, and straightedge, you will create a variety of geometric figures.

Chapter 1 | Tools of Geometry 1

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2 Chapter 1 | Tools of Geometry

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Mathematical Representations

Introduction

During this course, you will solve problems and work with many different representations

of mathematical concepts, ideas, and processes to better understand the world.

The following process icons are placed throughout the text.

Constructionl Use only a compass and a straightedge to create the geometric figure.

Discuss to Understandl Read the problem carefully.

l What is the context of the problem? Do you understand it?

l What is the question that you are being asked? Does it make sense?

Think for Yourselfl Do I need any additional information to answer the question?

l Is this problem similar to some other problem that I know?

l How can I represent the problem using a picture, a diagram, symbols, or some

other representation?

Work with Your Partnerl How did you do the problem?

l Show me your representation.

l This is the way I thought about the problem—how did you think about it?

l What else do we need to solve the problem?

l Does our reasoning and our answer make sense to one another?

Work with Your Groupl Show me your representation.

l This is the way I thought about the problem—how did you think about it?

l What else do we need to solve the problem?

l Does our reasoning and our answer make sense to one another?

l How can we explain our solution to one another? To the class?

Share with the Classl Here is our solution and how we solved it.

l We could only get this far with our solution. How can we finish?

l Could we have used a different strategy to solve the problem?

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Lesson 1.1 | Let's Get This Started 3

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Points, Lines, Planes, Rays, and Segments

OBJECTIVESIn this lesson you will:l Name points, lines, rays, line segments,

and planes.l Describe the intersections of lines and

planes.l Duplicate a line segment using a

compass and straightedge.l Add and subtract line segments.

KEY TERMSl point l coplanar linesl line l skew linesl collinear points l rayl plane l endpoint of a rayl compass l line segmentl straightedge l endpoints of a line segmentl sketch l congruent line segmentsl draw l duplicate a line segmentl construct

1.1

PROBLEM 1 Establishing a Common Language To communicate thoughts and ideas, a common language must be established.

Throughout this course, terms will be introduced and defined. However, this

section begins by introducing three terms that cannot be defined; they can only be

described. Three essential building blocks of geometry are point, line, and plane.

These three terms are called undefined terms; we can only describe and create

mathematical models to represent them.

A point is described as a location in space. A point has no size or shape but is often

represented using a dot.

Let's Get This Started

1


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A mathematical model of several points is shown. Each point represents the age and

height of a different child.

X

Y

C A

Age (years)

Hei

ght

(inch

es)

B D

E

1. Which point represents the oldest child?

2. Which points represent children of the same age?

3. Which points represent children of the same height?

4. What was used to name each point in Questions 1 through 3?

A line is described as a straight continuous arrangement of an infinite number of

points. A line has an infinite length, but no width. Arrowheads are used to indicate

that a line extends infinitely in opposite directions.

A mathematical model of several lines is shown.l

Em

n

p

C

A D

B

5. Does C determine a specific line in the model? Explain.


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6. Does CD determine a specific line in the model? Explain.

7. Does m determine a specific line in the model? Explain.

8. How many points are needed to determine a specific line?

9. What is an alternate name for line AB?

Line AB can be written using symbols as ‹

___ ›

AB and is read as “line AB.”

Collinear points are points that are located on the same line.

10. Name three points that are collinear in the model.

11. Name three points that are not collinear in the model.

A plane is described as a flat surface. A plane has an infinite length and width but no

depth. A plane extends infinitely in all directions. One model of a plane is the surface

of a still body of water. Planes are determined by three points, but are usually named

using one italic letter located near a corner of the plane as drawn. Three planes can

intersect in a variety of ways as shown.

pw

z

Figure 1

p

w

z

Figure 2

w

p

z

Figure 3

zp

w

Figure 4

p

w

z

Figure 5


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12. Describe the intersection of planes p, w, and z in each figure.

a. Figure 1

b. Figure 2

c. Figure 3

d. Figure 4

e. Figure 5

13. List all of the possible intersections of three planes.

14. Sketch and describe all possible intersections of a line and a plane.


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PROBLEM 2 Creating Geometric FiguresMany tools can be used to create geometric

figures. Some tools, such as a ruler or a

protractor, are classified as measuring tools.

A compass is a tool used to create arcs and

circles. A straightedge is a ruler with no

numbers. It is important to know when to use

each tool.

• When you sketch a geometric figure, the

figure is created without the use of tools.

• When you draw a geometric figure,

the figure is created with the use of

tools such as a ruler, straightedge, compass, or protractor. A drawing is more

accurate than a sketch.

• When you construct a geometric figure, the figure is created using only a

compass and a straightedge.

1. Create a geometric figure using collinear points A, B, and C such that point B is

located halfway between points A and C using each method. Label the points.

a. Sketch the figure.

b. Draw the figure.

2. Describe what you did differently to answer Questions 1(a) and 1(b).

Coplanar lines are two or more lines that are located in the same plane.

Skew lines are two or more lines that are not in the same plane.

3. Draw and label three coplanar lines.


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4. Look around your classroom. Describe the location of two skew lines.

A ray is a portion of a line that begins with a single point and extends infinitely in one

direction. The endpoint of a ray is the single point where the ray begins.

A ray is named using two capital letters, the first representing the endpoint and

the second representing any other point on the ray. Ray AB can be written using

symbols as

___

› AB and is read as “ray AB.”

1. Sketch and label

___

› AB .

2. Sketch and label

___

› BA .

3. Are

___

› AB and

___

› BA names for the same ray? Explain.

4. Use symbols to name the geometric figure shown.

F

G

PROBLEM 3 Using Undefined Terms to Define New Terms


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A line segment is a portion of a line that includes two points and all of the collinear

points between the two points. The endpoints of a line segment are the points

where the line segment begins and ends.

A line segment is named using two capital letters representing the two endpoints of

the line segment. Line segment AB can be written using symbols as ___

AB and is read

as “line segment AB.”

5. Draw and label ___

AB .

6. Draw and label ___

BA .

7. Are ___

AB and ___

BA names for the same line segment? Explain.

8. Use a ruler to measure ___

AB in Question 5.

9. The measure of ___

AB can be expressed in two different ways. Complete each

statement:

a. “AB � inches” is read as “the distance from point A to point B is

equal to inches.”

b. “m ___

AB � inches” is read as “the measure of line segment AB is equal to

inches.”

c. How do you read “m ___

CF � 3 inches”?

d. How do you read “SP � 8 inches”?

10. Use symbols to name the geometric figure.

F G

11. Use symbols to name the geometric figure.

F G


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If two line segments have equal measure, then the line segments have the same

length. Congruent line segments are two or more line segments of equal measure.

If m ___

AB � m ___

CD , then line segment AB is congruent to line segment CD by the

definition of congruent line segments. This statement can be written using symbols

as ___

AB � ___

CD and is read as “line segment AB is congruent to line segment CD.”

Use the congruence symbol, �, between congruent geometric figures and the equal

symbol, �, between references to lengths or distances.

12. Draw and label two congruent line segments.

Markers are used to indicate congruent segments in geometric figures. The following

figure shows ___

AB � ___

CD .

A DB C

13. Use symbols to write three valid conclusions based on the figure. How do you

read each conclusion?

F IG10 cm 10 cmH

14. Is there a difference between ‹

___

› HG and

‹

___

› GH ? Explain.

15. Is there a difference between

___

› GH and

___

› HG ? Explain.

16. Is there a difference between ____

HG and ____

GH ? Explain.


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17. Is there a difference between the statements JK � MN and ___

JK � ____

MN ? Explain.

A B

C

A B

C C

A B

D

1. Construct a line segment that is twice the length of line segment AB.

Draw a Starter Line

Use a straightedge

to draw a starter line

longer than segment

AB. Label point C on

the new segment.

Measure Length

Set your compass

at the length AB.

Copy Length

Place the compass

at C. Mark point D

on the new segment.

Duplicate Line SegmentsYou can duplicate a line segment by constructing an exact copy of the original.

PROBLEM 4 Construction


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2. Duplicate each line segment using a compass and a straightedge.

Y Z

U V W X

3. Use the line segments from Question 2 to construct a line segment with a

length that is equal to UV � YZ.

4. Use the line segments from Question 2 to construct a line segment with a

length that is equal to 2UV � WX � YZ.

Be prepared to share solutions and methods.

Lesson 1.2 | All About Angles 13

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1OBJECTIVESIn this lesson you will:l Name angles.l Classify angles.l Duplicate an angle using a

compass and straightedge.l Construct angle bisectors.

KEY TERMSl angle l obtuse anglel sides of an angle l straight anglel vertex of an angle l congruent anglesl protractor l duplicate an anglel degrees l bisectl acute angle l angle bisectorl right angle

PROBLEM 1 Naming AnglesAn angle is formed by two rays that share a common endpoint. The angle symbol

is �. The sides of an angle are the two rays. The vertex of an angle is the common

endpoint of the two rays.

An angle is shown with the sides and vertex labeled.

SideSide

Vertex

All About AnglesNaming Angles, Classifying Angles, Duplicating Angles, and Bisecting Angles

1

1.2


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1. Consider the angle shown.

A

B

C

a. Name the vertex of the angle.

b. Name the sides of the angle.

Use the diagram shown to answer Questions 2 through 10.

E

F G

DC

2. Does �C determine a specific angle in the diagram? Explain.

3. Does �E determine a specific angle in the diagram? Explain.

4. Does �ED determine a specific angle in the diagram? Explain.

5. Does �DEC determine a specific angle in the diagram? Explain.


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6. When can one capital letter be used to name an angle?

7. What is the difference between �FGE and �EGF?

8. What is the difference between �EFG and �EGF?

9. What are alternate names for �D?

10. How many letters are needed to name an angle?

11. Angles can also be named using numbers. Consider the diagram shown.

E1

2

FH

DC

G

a. What is an alternate name for �1?

b. What is an alternate name for �2?

12. Do �3 and �4 share a common side? Explain.

3

4

A

B


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A protractor is a basic tool used to measure angles. One unit of measure for angles

is degrees (º).

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11070

12060 130

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In this section, we will work with angles whose measures are greater than 0º and less

than or equal to 180º. Later in the course we will study circles and work with angles

whose measures are greater than 180º.

To measure an angle using a protractor, align the bottom of the protractor with one

side of the angle. Align the center of the protractor with the vertex of the angle.

The second side of the angle aligns with the angle’s degree measure. The correct

placement of the protractor is shown.

1. What is the measure of the angle shown?

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PROBLEM 2 Measuring Angles


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2. Is the measure of the angle shown 130º or 50º? Explain.

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12060 130

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3. How do you know when to use each of the two scales on a protractor?

4. Use the diagram shown to answer each question.

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AXW

R

a. What is the measure of �WAR?

b. What is the measure of �RAX?

c. What is the measure of �WAX?


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5. Use this diagram shown to determine the measure of each angle.

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EPT

S

R

Q

a. �SET b. �QEP

c. �REQ d. �REP

e. �TEQ f. �PES

g. �SER

6. Use a protractor to determine the measure of each angle to the nearest degree.

a. b.


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7. Which angle is larger? Explain.

8. Use a protractor to draw an angle with the given measure.

a. 30º angle b. 130º angle

An acute angle is an angle whose measure is greater than 0º, but less than 90º.

1. Draw and label an acute angle.

PROBLEM 3 Classifying Angles


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A right angle is an angle whose measure is equal to 90º. A square drawn at the

vertex is used to indicate a right angle in geometric figures.

2. Draw and label a right angle.

An obtuse angle is an angle whose measure is greater than 90º, but less than 180º.

3. Draw and label an obtuse angle.

A straight angle is an angle whose measure is equal to 180º. The sides of a straight

angle form a line.

4. Draw and label a straight angle.


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Congruent angles are two or more angles that have equal measures. An angle

congruence statement can be written using symbols as �A � �B and is read as

“angle A is congruent to angle B.”

5. Draw and label �A and �B such that �A � �B.

6. Use a protractor to measure �A and �B, then complete each statement.

a. “m�A � º is read as “the measure of angle A is equal to degrees.”

b. “m�B � º is read as “the measure of angle B is equal to degrees.”

c. How do you read “m�DEF � 110º?”

If m�A � m�B then �A is congruent to �B by the definition of congruent angles.

As with segments, use the congruent symbol, �, between the geometric figure

angles, and the equal symbol, �, between references to measures of angles.

Markers are used to indicate congruent angles in geometric figures. The following

figure shows �A � �B.

A B


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1You can duplicate an angle by constructing an exact copy of the original.

A

C

A

C C

A D

B

E

F

A D

B

C E

F

Draw a Starter Line

Use a straightedge

to draw a starter line.

Label point C on the

new segment.

Draw an Arc

Draw an arc with

center A. Using the

same radius, draw an

arc with center C.

Draw an Arc

Label points B, D,

and E. Draw an arc

with radius BD and

center E. Label the

intersection F.


Duplicate Angles

Draw a Ray

Draw ray CF.

�BAD � �FCE.


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1. Construct an angle that is twice the measure of �A.

A

To bisect means to divide into two equal parts.

1. If a line segment is bisected, what does that mean?

2. If an angle is bisected, what does that mean?


Angle Bisectors


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If a ray is drawn through the vertex of the angle and divides the angle into two angles

of equal measure or two congruent angles, the ray is called an angle bisector.

C B

A

C B

A

C B

DA

3. Construct the bisector of �A.

A

Draw an Arc

Place the compass at

C. Draw an arc that

intersects both sides

of the angle. Label

the intersections A

and B.

Draw an Arc

Place the compass at

A. Draw an arc, then

place the compass

point at B. Using the

same radius, draw

another arc.

Draw a Ray

Label the intersection

of the two arcs D. Use

a straightedge to draw

a ray through C and D.

Ray CD bisects �C.


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4. Construct an angle that is one fourth the measure of �H.

H

5. Describe how to construct an angle that is one eighth the measure of angle H.


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6. Use a compass and straightedge to show that the two angles formed by the

angle bisector of angle A are congruent. Describe each step.

A


Lesson 1.3 | Special Angles 27

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1OBJECTIVESIn this lesson you will:l Calculate the supplement of an angle.l Calculate the complement of an angle.l Construct a perpendicular. l Construct a perpendicular bisector.l Construct the midpoint of a segment.l Classify adjacent angles, linear pairs, and

vertical angles.

KEY TERMSl supplementary anglesl complementary anglesl perpendicularl midpoint of a segmentl segment bisectorl perpendicular bisectorl adjacent anglesl linear pairl vertical angles

PROBLEM 1 Supplements and ComplementsTwo angles are supplementary angles if the sum of their angle measures is equal

to 180º.

1. Use a protractor to draw a pair of supplementary angles that share a common

side, and then measure each angle.

2. Use a protractor to draw a pair of supplementary angles that do not share a

common side, and then measure each angle.

Special AnglesComplements, Supplements, Midpoints, Perpendiculars, and Perpendicular Bisectors

1

1.3


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3. Calculate the measure of an angle that is supplementary to �KJL.

J

K

22°

L

Two angles are complementary angles if the sum of their angle measures is equal

to 90º.

4. Use a protractor to draw a pair of complementary angles that share a common

side, and then measure each angle.

5. Use a protractor to draw a pair of complementary angles that do not share a

common side, and then measure each angle.

6. Calculate the measure of an angle that is complementary to �J.

62°

J


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7. Two angles are both congruent and supplementary. What is the measure of

each angle? Explain.

8. Two angles are both congruent and complementary. What is the measure of

each angle? Explain.

9. The complement of an angle is twice the measure of the angle. What is the

measure of each angle? Explain.

10. The supplement of an angle is half the measure of the angle. What is the

measure of each angle? Explain.


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Two lines, line segments, or rays are perpendicular if they intersect to

form 90º angles. The perpendicular symbol is �.

1. Name all angles that you know are right angles in the figure shown.

B

D

A

C

F

2. Draw ‹

___

› AB �

‹

___

› CD at point E. How many right angles are formed?

3. Draw ‹

___

› BC �

‹

___

› AB at point B. How many right angles are formed?

PROBLEM 2 Perpendicular Relationships


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1Perpendicular Line Through a Point on a Line

BC D

B

E

F

C D

Perpendicular Lines

B

F

C D

E

1. Construct a line perpendicular to the given line through point P.

P

Draw an Arc

Use B as the center

and draw an arc. Label

the intersections

points C and D.

Draw Arcs

Open the compass radius.

Use C and D as

centers and draw arcs

above and below the line.

Label the intersections

points E and F.

Draw a Line

Use a straightedge

to connect points E

and F. Line EF is

perpendicular to line CD.



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Perpendicular Line Through a Point Not on a Line

B

C D

B

E

F

C D

B

F

E

2. Construct a line perpendicular to ‹

___

› AG through point B.

B

A G

3. How is the construction of a perpendicular through a point on the line different

than the construction of a perpendicular through a point not on the line?

Draw an Arc

Use B as the center and

draw an arc. Label the

intersections points C

and D.

Draw Arcs

Open the compass

radius. Use C and D as

centers and draw arcs

above and below the line.

Label the intersections

points E and F.

Draw a Line

Use a straightedge

to connect points E

and F. Line EF is

perpendicular to line CD.


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1The midpoint of a segment is a point that divides the segment into two congruent

segments, or two segments of equal measure.

P M Q

___

PQ has midpoint M.

A segment bisector is a line, line segment, or ray that divides the line segment into

two line segments of equal measure, or two congruent line segments.

A perpendicular bisector is a line, line segment, or ray that intersects the midpoint

of a line segment at a 90 degree angle.

Perpendicular Bisector

A B

A

E

F

B

A

E

FB

Draw an Arc

Open the radius of

the compass to more

than half the length

of line segment AB.

Use endpoint A as the

center and draw an arc.

Draw an Arc

Keep the compass radius

and use point B as the

center and draw an arc.

Label the points formed

by the intersection of the

arcs point E and point F.

Draw a Line

Connect points E and F.

Line segment EF is the

perpendicular bisector of

line segment AB.


Midpoint and a Perpendicular Bisector


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1. Construct the perpendicular bisector of ___

FG . Label the perpendicular bisector

as ‹

___

› CD .

G

F

2. Label the point at which ‹

___

› CD intersects

___ FG as point E.

3. If ‹

___

› CD �

___ FG , what can you conclude?

4. If ‹

___

› CD bisects


5. If ‹

___

› CD is the perpendicular bisector of


6. Construct the midpoint of ___

PQ .

P Q


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PROBLEM 5 Adjacent Angles�1 and �2 are adjacent angles. �1 and �2 are not adjacent angles.

1

1 2

2

1 2

1

2

1. Describe adjacent angles.

2. Draw �2 so that it is adjacent to �1.

1

3. Is it possible to draw two angles that share a common vertex but do not share

a common side? If so, draw an example. If not, explain.


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4. Is it possible to draw two angles that share a common side, but do not

share a common vertex? If so, draw an example. If not, explain.

Adjacent angles are two angles that share a common vertex and share a

common side.

�1 and �2 form a linear pair. �1 and �2 do not form a linear pair.

1

2

1 2

1

1

2

2

1. Describe a linear pair of angles.

2. Draw �2 so that it forms a linear pair with �1.

1

3. Name all linear pairs in the figure shown.

1

2

3 4

PROBLEM 6 Linear Pairs


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4. If the angles that form a linear pair are congruent, what can you conclude?

A linear pair of angles are two adjacent angles that have noncommon sides that

form a line.

PROBLEM 7 Vertical Angles�1 and �2 are vertical angles. �1 and �2 are not vertical angles.

1

1

2

2

12

2 1

1. Describe vertical angles.

2. Draw �2 so that it forms a vertical angle with �1.

1


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3. Name all vertical angle pairs in the diagram shown.

1

2

3 4

4. Measure each angle in Question 3. What do you notice?

Vertical angles are two nonadjacent angles that are formed by two

intersecting lines.


Lesson 1.4 | A Little Dash of Logic 39

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1OBJECTIVESIn this lesson you will:l Define inductive reasoning and deductive

reasoning.l Identify methods of reasoning.l Compare and contrast methods of reasoning.l Create examples using inductive and deductive

reasoning.

KEY TERMSl inductive reasoningl deductive reasoning

1. Emma is watching her big sister do

homework. She notices the following:

• 42 � 4 � 4

• nine cubed is equal to nine times

nine times nine

• 10 to the fourth power is equal

to four factors of 10 multiplied

together

Emma concludes that raising a

number to a power is the same as

multiplying the number by itself as many times as indicated by the power.

How did Emma reach this conclusion?

2. Ricky read that exponents mean repeated multiplication. He had to enter seven

to the fourth power in a calculator but could not find the exponent button.

So, he entered 7 � 7 � 7 � 7 instead. How did Ricky reach this conclusion?

3. Contrast the reasoning used by Emma and Ricky.

A Little Dash of LogicTwo Methods of Logical Reasoning

PROBLEM 1

1

1.4


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4. Was Emma’s conclusion correct? Was Ricky’s conclusion correct?

5. Jennifer’s mother is a writing consultant. She was paid $900 for a ten-hour job

and $1980 for a twenty two-hour job.

a. How much does Jennifer’s mother charge per hour?

b. To answer Question 5(a), did you start with a general rule and make a

conclusion or did you start with specific information and create a

general rule?

6. Your friend Aaron tutors elementary school students. He tells you that the job

pays $8.25 per hour.

a. How much does Aaron earn if he works 4 hours?

b. To answer Question 6(a), did you start with a general rule and make a

conclusion or did you start with specific information and create a

general rule?

PROBLEM 2The ability to use information to reason and make conclusions is very important in

life and in mathematics. This lesson focuses on two methods of reasoning. You can

construct the vocabulary for each type of reasoning by thinking about what prefixes,

root words, and suffixes mean.

Look at the following information. Remember that a prefix is a beginning of a word.

A suffix is an ending of a word.

• in—a prefix that can mean toward or up to

• de—a prefix that can mean down from

• duc— a base or root word meaning to lead and often to think, from the Latin

word duco

• -tion—a suffix that forms a noun meaning the act of

1. Form a word that means “the act of thinking down from.”

2. Form a word that means “the act of thinking toward or up to.”


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Inductive reasoning is reasoning that involves using specific examples to make a

conclusion. Many times in life you must make generalizations about observations

or patterns and apply these generalizations to unfamiliar situations. For example,

you learn how to ride a bike by falling down, getting back up, and trying again.

Eventually, you are able to balance on your own. After learning to ride your own bike,

you can apply that knowledge and experience to ride an unfamiliar bike.

Deductive reasoning is reasoning that involves using a general rule to make a

conclusion. For example, you learn the rule for which direction to turn a screwdriver:

“righty tighty, lefty loosey.” If you want to unscrew a screw, you apply the rule and

turn counterclockwise.

3. Look back at Problem 1. Who used inductive reasoning?

4. Who used deductive reasoning?

PROBLEM 3

1. Your best friend reads a newspaper article that

states that use of tobacco greatly increases

the risk of cancer. He then notices that his

neighbor Matilda smokes. He is concerned

that Matilda has a high risk of cancer.

a. What is the specific information in

this problem?

b. What is the general information in

this problem?

c. What is the conclusion in this problem?

d. Did your friend use inductive or deductive reasoning to make the

conclusion? Explain.


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e. Is your friend’s conclusion correct? Explain.

2. Molly returned from a trip to London and tells you, “It rains every day in

England!” She explains that it rained each of the five days she was there.

a. What is the specific information in this problem?

b. What is the general information in this problem?

c. What is the conclusion in this problem?

d. Did Molly use inductive or deductive reasoning to make the conclusion?

Explain your reasoning.

e. Is Molly’s conclusion correct? Explain.

3. You take detailed notes in history class and math class. A classmate is going

to miss biology class tomorrow to attend a field trip. His biology teacher asks

him if he knows someone in class who always takes detailed notes. He gives

your name to the teacher. The biology teacher suggests he borrow your biology

notes because he concludes that they will be detailed.

a. What conclusion did your classmate make? Why?

b. What type of reasoning did your classmate use? Explain.

c. What conclusion did the biology teacher make? Why?

d. What type of reasoning did the biology teacher use? Explain.


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e. Will your classmate’s conclusion always be true? Will the biology teacher’s

conclusion always be true? Explain.

4. The first four numbers in a sequence are 4, 15, 26, and 37.

a. What is the next number in the sequence? How did you calculate the

next number?

b. What types of reasoning did you use and in what order to make the

conclusion?

5. Write a short note to a friend explaining induction and deduction. Include

definitions of both terms and examples that are very easy to understand.


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There are two reasons why a conclusion may be false. Either the assumed

information is false or the argument is not valid.

1. Derek tells his little brother that it will not rain for the next thirty days because

he “knows everything.” Why is this conclusion false?

2. Two lines are not parallel so the lines must intersect. Why is this

conclusion false?

3. Write an example of a conclusion that is false because the assumed

information is false.

4. Write an example of a conclusion that is false because the argument

is not valid.

Be prepared to share your solutions and methods.

PROBLEM 4

Lesson 1.5 | Conditionals 45

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1OBJECTIVESIn this lesson you will:l Identify the hypothesis and

conclusion of a conditional statement.

l Explore the truth value of conditional statements.

l Use a truth table.l Differentiate between postulates

and theorems.l Differentiate between Euclidean

and non-Euclidean geometries.

KEY TERMSl conditional statement l postulatel propositional form l theoreml propositional variables l Euclidean geometryl hypothesis l Linear Pair Postulatel conclusion l Segment Addition Postulatel truth value l Angle Addition Postulatel truth table

A conditional statement is a statement that can be written in the form “If p, then q.”

This form is the propositional form of a conditional statement. It can also be written

using symbols as p → q and is read as “p implies q.” The variables p and q are

propositional variables. The hypothesis of a conditional statement is the variable p.

The conclusion of a conditional statement is the variable q.

The truth value of a conditional statement is whether the statement is true or false.

If a conditional statement could be true, then the truth value of the statement is

considered true. The truth value of a conditional statement is either true or false, but

not both.

ConditionalsConditional Statements, Postulates, and Theorems

1

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PROBLEM 1 Conditional Statements and Truth Values

Consider the following conditional statement: If the measure of an angle is 32º, then

the angle is acute.

1. What is the hypothesis p?

2. What is the conclusion q?

3. If p is true and q is true, then the truth value of a conditional statement is true.

a. What does the phrase “If p is true” mean in terms of the conditional

statement?

b. What does the phrase “If q is true” mean in terms of the conditional

statement?

c. Explain why the truth value of the conditional statement is true.

4. If p is true and q is false, then the truth value of a conditional statement

is false.

a. What does the phrase “If p is true” mean in terms of the conditional

statement?

b. What does the phrase “If q is false” mean in terms of the conditional

statement?

c. Explain why the truth value of the conditional statement is false.


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5. If p is false and q is true, then the truth value of a conditional statement is true.

a. What does the phrase “If p is false” mean in terms of the conditional

statement?

b. What does the phrase “If q is true” mean in terms of the conditional

statement?


6. If p is false and q is false, then the truth value of a conditional statement is true.

a. What does the phrase “If p is false” mean in terms of the conditional

statement?

b. What does the phrase “If q is false” mean in terms of the conditional

statement?


A truth table is a table that summarizes all possible truth values for a conditional

statement p → q. The first two columns of a truth table represent all possible truth

values for the propositional variables p and q. The last column represents the truth

value of the conditional statement p → q.

The truth value for the conditional statement “If the measure of an angle is 32º, then

the angle is acute“ is shown.

p q p → q

T T T

T F F

F T T

F F T


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7. Consider the conditional statement “If m ___

AB � 6 inches and m ___

BC � 6 inches,

then ___

AB � ___

BC .”

a. What is the hypothesis p?

b. What is the conclusion q?

c. If both p and q are true, what does that mean? What is the truth value of the

conditional statement if both p and q are true?

d. If p is true and q is false, what does that mean? What is the truth value of

the conditional statement if p is true and q is false?

e. If p is false and q is true, what does that mean? What is the truth value of

the conditional statement if p is false and q is true?

f. If both p and q are false, what does that mean? What is the truth value of the

conditional statement if both p and q are false?


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For each conditional statement, draw a diagram and then write the hypothesis as the

“Given” and the conclusion as the “Prove.”

1. If

___

› BD bisects �ABC, then �ABD � �CBD.

Given:

Prove:

2. ____

AM � ____

MB , if M is the midpoint of ___

AB .

Given:

Prove:

PROBLEM 2 Rewriting Conditional Statements


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3. If ‹

___

› AB �

___

› CD at point C, then �ACD is a right angle and �BCD is a right angle.

Given:

Prove:

4. m�DEG � m�GEF � 180º, if �DEG and �GEF are a linear pair.

Given:

Prove:

5.

___

› WX is the perpendicular bisector of

___ PR , if

___

› WX �

___ PR and

___

› WX bisects

___ PR .

Given:

Prove:


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6. If �ABD and �DBC are complementary then

___

› BA �

___

› BC .

Given:

Prove:

PROBLEM 3 Postulates and TheoremsA postulate is a statement that is accepted

without proof.

A theorem is a statement that can

be proven.

The Elements is a book written by the

Greek mathematician Euclid. He used

a small number of undefined terms and

postulates to systematically prove many

theorems. As a result, Euclid was able to

develop a complete system we now know as

Euclidean geometry.

Euclid’s first five postulates are:

1. A straight line segment can be drawn

joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as

radius and one endpoint as center.

4. All right angles are congruent.

5. If two lines are drawn that intersect a third line in such a way that the sum of

the inner angles on one side is less than two right angles, then the two lines

inevitably must intersect each other on that side if extended far enough.

(This postulate is equivalent to what is known as the parallel postulate.)


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Euclid used only the first four postulates to prove the first 28 propositions or

theorems of The Elements, but was forced to use the fifth postulate, the parallel

postulate, to prove the 29th theorem.

The Elements also includes five “common notions”:

1. Things that equal the same thing also equal one another.

2. If equals are added to equals, then the wholes are equal.

3. If equals are subtracted from equals, then the remainders are equal.

4. Things that coincide with one another equal one another.

5. The whole is greater than the part.

It is important to note that Euclidean geometry is not the only system of geometry.

Examples of non-Euclidian geometries include hyperbolic and elliptic geometry. The

essential difference between Euclidean and non-Euclidean geometry is the nature of

parallel lines.

Another way to describe the differences between these geometries is to consider

two lines in a plane that are both perpendicular to a third line.

Euclidean

Hyperbolic

Non-Euclidean

Elliptic

• In Euclidean geometry, the lines remain at a constant distance from each other

and are known as parallels.

• In hyperbolic geometry, the lines “curve away” from each other.

• In elliptic geometry, the lines “curve toward” each other and eventually

intersect.

Using this textbook as a guide, you will develop your own system of geometry, just

like Euclid. You already used the three undefined terms point, line, and plane to

define related terms such as line segment and angle.

Your journey continues with the introduction of three further fundamental postulates:

• The Linear Pair Postulate

• The Segment Addition Postulate

• The Angle Addition Postulate

You will use these postulates to make various conjectures. If you are able to prove

our conjectures, then the conjecture will become a theorem. The theorem can then

be used to make even more conjectures, which may also become theorems, thus

defining your own system of geometry.


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The Linear Pair Postulate states: “If two angles are a linear pair, then the angles

are supplementary.”

1. Sketch and label a linear pair.

2. Use your sketch and the Linear Pair Postulate to complete the statement.

� and � are .

3. Use the definition of supplementary angles to complete the statement.

m� � m� �

The Segment Addition Postulate states: “If point B is on ___

AC and between points A

and C, then AB � BC � AC.”

4. Sketch and label collinear points D, E, and F with point E between points D

and F.

5. Use your sketch and the Segment Addition Postulate to complete the

statement.

� �

6. Use your answer to Question 5 to complete the statement.

m � m � m


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The Angle Addition Postulate states: “If point D lies in the interior of �ABC, then

m�ABD � m�DBC � m�ABC.”

7. Sketch and label �DEF with

___

› EG drawn in the interior of �DEF.

8. Use your sketch and the Angle Addition Postulate to complete the statement.

m� � m� � m�


Lesson 1.6 | Forms of Proof 55

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Paragraph Proof, Two-Column Proof, Construction Proof, and Flow Chart Proof

OBJECTIVESIn this lesson you will:l Use the addition and subtraction

properties of equality.l Use the reflexive, substitution,

and transitive properties.l Write a paragraph proof.l Complete a two-column proof.l Perform a construction proof.l Complete a flow chart proof.

KEY TERMSl Addition Property

of Equalityl Subtraction Property

of Equalityl Reflexive Propertyl Substitution Propertyl Transitive Propertyl paragraph proofl two-column proof

PROBLEM 1 Properties of Real Numbers in GeometryMany properties of real numbers can be applied in geometry. These properties are

important when making conjectures and proving new theorems.

The Addition Property of Equality states: “If a, b, and c are real numbers and

a � b, then a � c � b � c.”

The Addition Property of Equality can be applied to angle measures, segment

measures, and distances.

• Angle measures: If m�1 � m�2, then m�1 � m�3 � m�2 � m�3.

• Segment measures: If m ___

AB � m ___

CD , then m ___

AB � m ___

EF � m ___

CD � m ___

EF .

• Distances: If AB � CD, then AB � EF � CD � EF.

1. Write a statement that applies the Addition Property of Equality to angles.

2. Write a statement that applies the Addition Property of Equality to segments.

Forms of Proof

l construction proofl flow chart proofl Right Angle Congruence

Theoreml Congruent Supplement

Theoreml Congruent Complement

Theoreml Vertical Angle Theorem

1

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The Subtraction Property of Equality states: “If a, b, and c are real numbers and

a � b, then a � c � b � c.”

The Subtraction Property of Equality can be applied to angle measures, segment

measures, and distances.

• Angle measures: If m�1 � m�2, then m�1 � m�3 � m�2 � m�3.


AB � m ___

CD , then m ____

AB � m ___

EF � m ___

CD � m ___

EF .

• Distances: If AB � CD, then AB � EF � CD � EF.

3. Write a statement that applies the Subtraction Property of Equality to angles.

4. Write a statement that applies the Subtraction Property of Equality to

segments.

The Reflexive Property states: “If a is a real number, then a � a.”

The Reflexive Property can be applied to angle measures, segment measures,

distances, congruent angles, and congruent segments.

• Angle measures: m�1 � m�1

• Segment measures: m ___

AB � m ___

AB

• Distances: AB � AB

• Congruent angles: �1 � �1

• Congruent segments: ___

AB � ___

AB

5. Write a statement that applies the Reflexive Property to angles.

6. Write a statement that applies the Reflexive Property to segments.

The Substitution Property states: “If a and b are real numbers and a � b, then a

can be substituted for b.”

The Substitution Property can be applied to angle measures, segment measures,

and distances.

• Angle measures: If m�1 � 56° and m�2 � 56°, then m�1 � m�2.


AB � 4 mm and m ___

CD � 4 mm, then m ___

AB � m ___

CD .

• Distances: If AB � 12 ft and CD � 12 ft, then AB � CD.


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7. Write a statement that applies the Substitution Property to angles.

8. Write a statement that applies the Substitution Property to segments.

The Transitive Property states: “If a, b, and c are real numbers, a � b, and b � c,

then a � c.”

The Transitive Property can be applied to angle measures, segment measures,

distances, congruent angles, and congruent segments.

• Angle measures: If m�1 � m�2 and m�2 � m�3, then m�1 � m�3.


AB � m ___

CD and m ___

CD � m ___

EF , then m ___

AB � m ___

EF .

• Distances: If AB � CD and CD � EF, then AB � EF.

• Congruent angles: If �1 � �2 and �2 � �3, then �1 � �3.

• Congruent segments: If ___

AB � ___

CD and ___

CD � ___

EF , then ___

AB � ___

EF .

9. Write a statement that applies the Transitive Property to angles.

10. Write a statement that applies the Transitive Property to congruent segments.

PROBLEM 2 Various Forms of Proof

A conditional statement is true if it can be proven to be true. A proof can be

presented in many different forms, including:

• A paragraph proof is a proof that the steps and corresponding reasons are

written in complete sentences.

• A two-column proof is a proof in which the steps are written in the left column

and the corresponding reasons in the right column. Each step is numbered and

the same number is used for the corresponding reason.

• A construction proof is a proof that results from creating an object with

specific properties using only a compass and straightedge.

• A flow chart proof is a proof in which the steps and corresponding reasons

are written in boxes. Arrows connect the boxes and indicate how each step

and reason is generated from one or more other steps and reasons.


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In this section, you will use each of these forms of proof. You may find some forms

of proof to be easier or shorter but you should be familiar with all four forms.

1. Draw four collinear points. Label the points A, B, C, and D such that point B

lies between points A and C, point C lies between points B and D, and

___

AB � ___

CD .

2. Consider the conditional statement: If ___

AB � ___

CD , then ___

AC � ___

BD . Write the

hypothesis as the “Given” and the conclusion as the “Prove.”

Given:

Prove:

3. Complete the flow chart proof of the conditional statement in Question 2 by

writing the reason for each statement in the boxes provided.

Given:

Prove:

AB CD

m AB m CD

m AB m BC m CD m BC

m AC m BD

m BC m BC

m AB m BC m AC

m BC m CD m BD

AC BD

� � �

�


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4. Create a two-column proof of the conditional statement in Question 2.

Each box of the flow chart proof in Question 3 should appear as a row in

the two-column proof.

Given:

Prove:

Statements Reasons

5. Write a paragraph proof of the conditional statement in Question 2. Each row

of the two-column proof in Question 4 should appear as a sentence in the

paragraph proof.

6. Create a proof by construction of the conditional statement in Question 2.

A B C D

Given:

Prove:


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PROBLEM 3 Proof of the Right Angle Congruence Theorem

The Right Angle Congruence Theorem states: “All right angles are congruent.”

A BC

D

Given: �ACD and �BCD are right angles.

Prove: �ACD � �BCD

Complete the flow chart of the Right Angle Congruence Theorem by writing the

statement for each reason in the boxes provided.

GivenGiven

Definition of right angles

Definition of right anglesTransitive Property of Equality

Definition of right anglesDefinition of congruent angles

Definition of right angles


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The Congruent Supplement Theorem states: “If two angles are supplements of the

same angle or of congruent angles, then the angles are congruent.”

1 2 3 4

1. Use the diagram to write the “Given” statements for the Congruent Supplement

Theorem. The “Prove” statement is provided.

Given:

Given:

Given:

Prove: �1 � �3

2. Complete a flow chart proof of the Congruent Supplement Theorem by drawing

arrows to connect the steps in a logical sequence.

PROBLEM 4 Proofs of the Congruent Supplement Theorem

�1 is supplementary to �2Given

�2 � �4Given

m�1 � m�2 � 180° Definition of supplementary angles

�3 is supplementary to�4Given

m�2 � m�4 Definition of congruent angles

m�1 � m�3 Subtraction Property of Equality

�1 � �3 Definition of congruent angles

m�1 � m�2 � m�3 � m�4Substitution Property

m�3 � m�4 � 180° Definition of supplementary angles


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3. Create a two-column proof of the Congruent Supplement Theorem.

Each box of the flow chart proof in Question 2 should appear as a row in the

two-column proof.

Statements Reasons

PROBLEM 5 Proofs of the Congruent Complement Theorem

The Congruent Complement Theorem states: “If two angles are complements of

the same angle or of congruent angles, then they are congruent.”

1. Draw and label a diagram illustrating this theorem.

2. Use the diagram to write the “Given” and “Prove” statements for the Congruent

Complement Theorem.

Given:

Given:

Given:

Prove:


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3. Create a flow chart proof of the Congruent Complement Theorem.

4. Create a two-column proof of the Congruent Complement Theorem.

Each box of the flow chart proof in Question 3 should appear as a row in

the two-column proof.

Statements Reasons


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1. The Vertical Angle Theorem states: “Vertical

angles are congruent.”

1 2

34

2. Use the diagram to write the “Prove” statements for the Vertical Angle

Theorem. The “Given” statements are provided.

Given: �1 and �2 are a linear pair




Prove:

Prove:

3. Create a flow chart proof of the first “Prove” statement of the Vertical

Angle Theorem.

PROBLEM 6 Proofs of the Vertical Angle Theorem


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4. Create a two-column proof of the second “Prove” statement of the Vertical

Angle Theorem.

Given:

Given:

Prove:

Statements Reasons


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Given: �DEG � �HEF

Prove: �DEH � �GEF

D H

E F

G

1. Prove the conditional statement using either a paragraph proof or a

two-column proof.

PROBLEM 7 Proofs using the Angle Addition Postulate


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1. Which form of proof is easiest to understand? Hardest to understand? Explain.

2. Which form of proof is easiest to write? Hardest to write? Explain.

3. Which form of proof has the fewest steps? The most steps? Explain.

4. Which form of proof do you prefer? Explain.

Once a theorem has been proven, it can be used as a reason in another proof. Using

theorems that have already been proven allows you to write shorter proofs.

In this chapter, you proved the following theorems:

• The Right Angle Congruence Theorem: All right angles are congruent.

• The Congruent Supplement Theorem: Supplements of congruent angles, or of

the same angle are congruent.

• The Congruent Complement Theorem: Complements of congruent angles, or of

the same angle are congruent.

• The Vertical Angle Theorem: Vertical angles are congruent.

A list of theorems that you prove throughout this course will be an excellent resource

as you continue to make new conjectures and expand our system of geometry.

Be prepared to share your solutions and methods.

PROBLEM 8 Summary


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Chapter 1 Checklist

KEY TERMSl point (1.1)l line (1.1)l collinear points (1.1)l plane (1.1)l compass (1.1)l straightedge (1.1)l sketch (1.1)l draw (1.1)l construct (1.1)l coplanar lines (1.1)l skew lines (1.1)l ray (1.1)l endpoint of a ray (1.1)l line segment (1.1)l endpoints of a line

segment (1.1)l congruent line

segments (1.1)l duplicate a line

segment (1.1)l angle (1.2)l sides of an angle (1.2)

l vertex of an angle (1.2)l protractor (1.2)l degrees (1.2)l acute angle (1.2)l right angle (1.2)l obtuse angle (1.2)l straight angle (1.2)l congruent angles (1.2)l duplicate an angle (1.2)l bisect (1.2)l angle bisector (1.2)l supplementary angles (1.3)l complementary angles (1.3)l perpendicular (1.3)l midpoint of a segment (1.3)l segment bisector (1.3)l perpendicular bisector (1.3)l adjacent angles (1.3)l linear pair (1.3)l vertical angles (1.3)l inductive reasoning (1.4)l deductive reasoning (1.4)

l conditional statement (1.5)l propositional form (1.5)l propositional variables (1.5)l hypothesis (1.5)l conclusion (1.5)l truth value (1.5)l truth table (1.5)l postulate (1.5)l theorem (1.5)l Euclidean geometry (1.5)l Addition Property of

Equality (1.6)l Subtraction Property of

Equality (1.6)l Reflexive Property (1.6)l Substitution Property (1.6)l Transitive Property (1.6)l paragraph proof (1.6)l two-column proof (1.6)l construction proof (1.6)l flow chart proof (1.6)

POSTULATESl Linear Pair Postulate (1.5) l Segment Addition

Postulate (1.5)l Angle Addition Postulate (1.5)

THEOREMSl Right Angle Congruence

Theorem (1.6)l Congruent Supplement

Theorem (1.6)

l Congruent Complement Theorem (1.6)

l Vertical Angle Theorem (1.6)

CONSTRUCTIONSl line segments (1.1)l angles (1.2)

l angle bisectors (1.2)l perpendicular lines (1.3)

l midpoint (1.3)

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Chapter 1 | Checklist 69

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Identifying Points, Lines, and Planes

A point is a location in space that has no size or shape. A line is a straight

continuous arrangement of an infinite number of points. A plane is a flat surface that

has an infinite length and width, but no depth. Collinear points are points that are

located on the same line. Coplanar lines are two or more lines that are located in the

same plane. Skew lines are two or more lines that are not in the same plane.

Examples:

p

qC E

D F

A B

Points A and B lie on line AB, points C and D lie on line CD, and points E and F lie on

line EF.

Line AB lies in plane q. Lines CD and EF lie in plane p.

Points A and B are collinear. Points C and D are collinear. Points E and F are

collinear.

Lines CD and EF are coplanar.

Lines AB and CD are skew. Lines AB and EF are skew.

Planes p and q intersect.

Determining Measures of Angles

To determine the measure of an angle, use a protractor. Align the center of

the protractor with the vertex of the angle. Each line on the protractor has two

measures — one for an acute angle and one for an obtuse angle.

Example:

90 10080

11070

12060 130

50 14040

15030

16020

17010

80

10070

11060

12050

130

4014

0

3015

0

2016

0

10 170

BDA

C

The measure of angle ABC is 140 degrees. The measure of angle DBC is 40 degrees.

1.1

1.2


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Classifying Angles as Acute, Obtuse, Right, or Straight

An acute angle is an angle whose measure is greater than 0 degrees, but less than

90 degrees. An obtuse angle is an angle whose measure is greater than 90 degrees,

but less than 180 degrees. A right angle is an angle whose measure is equal to

90 degrees. A straight angle is an angle whose measure is equal to 180 degrees.

Examples:

V W

ZY

X

Angles YWZ and ZWX are acute angles.

Angle VWZ is an obtuse angle.

Angles VWY and XWY are right angles.

Angle VWX is a straight angle.

Identifying Complementary and Supplementary Angles

Two angles are supplementary if the sum of their measures is 180 degrees.

Two angles are complementary if the sum of their measures is 90 degrees.

Examples:In the diagram above, angles YWZ and ZWX are complementary angles.

In the diagram above, angles VWY and XWY are supplementary angles.

Also, angles VWZ and XWZ are supplementary angles.

Identifying Linear Pairs and Vertical Angles

A linear pair of angles consists of two adjacent angles that have noncommon

sides that form a line. Vertical angles are nonadjacent angles formed by two

intersecting lines.

Examples:

m

n

12

34

Angles 1 and 2 form a linear pair. Angles 2 and 3 form a linear pair. Angles 3 and

4 form a linear pair. Angles 4 and 1 form a linear pair.

Angles 1 and 3 are vertical angles. Angles 2 and 4 are vertical angles.

1.2

1.3

1.3


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Identifying Types of Reasoning

Inductive reasoning is reasoning that involves using specific examples to make a

conclusion. Deductive reasoning is reasoning that involves using a general rule to

make a conclusion.

Examples:Michael notices that every Wednesday morning, a garbage truck comes to his

neighborhood. He determines that Wednesday mornings must be garbage collection

day. He used inductive reasoning because he used his specific experiences to make

a general conclusion.

Suzanne read in the city newspaper that Wednesday mornings are garbage

collection day in her neighborhood. So, each Tuesday night she takes her garbage

to the curb to be picked up the next morning. Suzanne used deductive reasoning

because she used a rule about garbage collection in her neighborhood to make

a conclusion.

Rewriting Conditional Statements

A conditional statement is a statement that can be written in the form “If p, then

q.” The hypothesis of a conditional statement is the variable p. The conclusion of a

conditional statement is the variable q.

Examples:Consider the following statement: If two angles form a linear pair, then the sum of the

measures of the angles is 180 degrees. The statement is a conditional statement.

The hypothesis is “two angles form a linear pair,” and the conclusion is “the sum of the

measures of the angles is 180 degrees.” The conditional statement can be rewritten with

the hypothesis as the “Given” statement and the conclusion as the “Prove” statement.

Given: Two angles form a linear pair.

Prove: The sum of the measures of the angles is 180 degrees.

1.4

1.5


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Determining the Difference Between Euclidean and Non-Euclidean Geometry

Euclidean geometry is a system of geometry developed by the Greek mathematician

Euclid that included the following five postulates.

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as a

radius and one point as the center.

4. All right angles are congruent.

5. If two lines are drawn that intersect a third line in such a way that the sum of

the inner angles on one side is less than two right angles, then the two lines

inevitably must intersect each other on that side if extended far enough.

Examples:

Euclidean geometry: Non-Euclidean geometry:

Using the Linear Pair Postulate

The Linear Pair Postulate states: “If two angles are a linear pair, then the angles

are supplementary.”

Example:

P Q S

R

38°

m�PQR � m�SQR � 180º

38º � m�SQR � 180º

m�SQR � 180º � 38º

m�SQR � 142º

1.5

1.5


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Using the Segment Addition Postulate

The Segment Postulate states: “If point B is on segment AC and between

points A and C, then AB � BC � AC.”

Example:

4 m 10 m

A B C

AB � BC � AC

4 � 10 � AC

AC � 14 m

Using the Angle Addition Postulate

The Angle Addition Postulate states: “If point D lies in the interior of angle ABC, then

m�ABD � m�DBC � m�ABC.”

Example:

A B

D

C

24°

39°

m�ABD � m�DBC � m�ABC

24º � 39º � m�ABC

m�ABC � 63º

1.5

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Using Properties of Real Numbers in Geometry

The Addition Property of Equality states: “If a, b, and c are real numbers and a � b,

then a � c � b � c.”

The Subtraction Property of Equality states: “If a, b, and c are real numbers and

a � b, then a � c � b � c.”

The Reflexive Property states: “If a is a real number, then a � a.”

The Substitution Property states: “If a and b are real numbers and a � b, then a can

be substituted for b.

The Transitive Property states: “If a, b, and c are real numbers and a � b and b � c,

then a � c.”

Examples:

Addition Property of Equality applied to angle measures: If m�1 � m�2, then

m�1 � m�3 � m�2 � m�3.

Subtraction Property of Equality applied to segment measures: If m ___

AB � m ___

CD , then

m ___

AB � m ___

EF � m ___

CD � m ___

EF .

Reflexive Property applied to distances: AB � AB

Substitution Property applied to angle measures: If m�1 � 20º and m�2 � 20º, then

m�1 � m�2.

Transitive Property applied to segment measures: If m ___

AB � m ___

CD and m ___

CD � m ___

EF ,

then m ___

AB � m ___

EF .

Using the Right Angle Congruence Theorem

The Right Angle Congruence Theorem states: “All right angles are congruent.”

Example:

F

G

KH

J

�FJH � �GJK

1.6

1.6


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Using the Congruent Supplement Theorem

The Congruent Supplement Theorem states: “If two angles are supplements of the


Example:

Y Z

XV

W

�VWZ � �XWY

Using the Congruent Complement Theorem

The Congruent Complement Theorem states: “If two angles are complements of the


Example:

1

23

4

�2 � �4

Using the Vertical Angle Theorem

The Vertical Angle Theorem states: “Vertical angles are congruent.”

Example:

12

34

s

t

�1 � �3 and �2 � �4

1.6

1.6

1.6


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