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Name _____________________________ Period ______

GEOMETRY – Chapter One – BASICS OF GEOMETRY Geometry, like much of mathematics and science, developed when people began recognizing and describing

patterns. In this course, you will study many amazing patterns that were discovered by people throughout

history and all around the world.

DISCOVERING GEOMETRY Over 6000 years ago, geometry consisted primarily of practical rules for measuring land and for

building. In fact, the term geometry means “____________________________________________________.”

The rules were found by observation and experimentation. Some early geometric methods were very accurate.

For example, the sides of the Great Pyramid at Gizeh were accurate to 1 centimeter in 180 meters (1/18,000 cm).

A more formal study of geometry began with the Greeks. About 600 B.C. a Greek mathematician named

Thales made a number of geometric conjectures with logical arguments. He used abstract diagrams to explore

possible geometric relationships. Follow the steps below to find one of the results that Thales discovered.

1. Mark a point on the semicircle. Connect the point to the

ends of the semicircle with a ruler. 2. Choose another point on the semicircle and

follow the same procedure.

3. Repeat one more time.

4. What did Thales notice about the

angles formed in each case?

___________________________________________

________________________________________________

ORGANIZING GEOMETRY Until the time of Euclid (300 B.C.), geometry was a collection of ideas. Euclid organized this information into

a system. He began with a few basic postulates,, or statements that are accepted as true. Then, reasoning

logically from his postulates, he was able to prove a number theorems, or statements that are proven to be true.

He published a 13 volume work called The Elements. In his systematic approach, figures are constructed using

only a compass and a straightedge. Since Euclid’s time The Elements has been translated into more languages

and published in more editions than any other book except the Bible.

USING GEOMETRY Geometric ideas are applied in many ways. We are surrounded by objects with pleasing or useful shapes. We

see arcs of circles in rainbows, hexagons in honeycombs, cubes in salt crystals, and spheres in soap bubbles.

Architects have made use of a wide variety of geometric shapes. You will learn to recognize and describe

patterns of your own. Sometimes, patterns allow you to make accurate predictions.

“As you embark on your study of geometry, you may wonder how successful you will be in your efforts.

Because geometry is a logical system, it is necessary to spend time on a regular basis to master the basic ideas

contained within it. A positive attitude and a willingness to try to do your best are also important factors in

determining how you will do. Pythagoras was a Greek geometer who lived about 2500 years ago.. He wondered whether he could teach geometry even to a reluctant student.

After finding such a student, Pythagoras agreed to pay him an obel for each theorem that he learned. Because the student was very poor, he worked

diligently. After a time, however, the student realized that he had become more interested in geometry than in the money he was accumulating. In fact, he became so intrigued with his studies that he begged Pythagoras to go faster, now offering to pay him back an obel for each new theorem. Eventually,

Pythagoras got all of his money back!” (Harold R. Jacobs)

I hope you will find geometry both enjoyable and rewarding. I look forward to this school year, for the

memories we will make and for the knowledge that we will learn.

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Section 1.1 -- Patterns and Inductive Reasoning

GOAL 1: Finding and Describing Patterns Ex. 1: Sketch the next figure in the pattern.

Ex. 2: Sketch the next figure in the pattern.

Ex. 3: Describe a pattern in the sequence of numbers. Predict the next two numbers.

a. 2, 5, 8, 11, _____, _____ b. 27, 9, 3, 1, _____, _____ c. 2, 5, 11, 23, _____, _____

___________________________________________________________________________.

Describe ______________________ ________________________ _____________________

GOAL 2: Using Inductive Reasoning Vocabulary: Vocabulary words are highlighted in yellow in this book.

A conjecture _________________________________________________________________________.

Inductive Reasoning____________________________________________________________________.

A counterexample______________________________________________________________________.

Much of the reasoning in geometry consists of three stages.

1. Look for a Pattern Look at several examples. Use diagrams and tables to help discover a pattern.

2. Make a Conjecture Use the examples to make a general conjecture.

3. Verify the Conjecture Use logical reasoning to verify that the conjecture is true in all cases.

Ex. 5 Complete the conjecture based on the pattern you observe.

3 ∙ 8 = 24 6 ∙ 5 = 30 9 ∙ 12 = 108 11 ∙ 24 = 264 102 ∙ 31 = 3162

The product of an odd number and an even number is

______________________________________.

Ex. 6 Complete the conjecture.

Conjecture: The sum of the first n odd positive integers is __________________?

To show that a conjecture is false, you only need to find one counterexample.

Ex. 7 Show the conjecture is false by finding a counterexample.

a. The difference of two whole numbers is a whole number.

b. All odd numbers are prime.

Not every conjecture is known to be true or false. These are called unproven or undecided.

Ex. 8 (Example 5 from text) In the early 1700s a Prussian mathematician named Goldbach noticed that many

even numbers greater than 2 can be written as the sum of two primes. Show this is true for even #s 20 to 30.

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Section 1.2 Points, Lines, and Planes

GOAL 1 Using Undefined Terms and Definitions A definition uses known words to describe a new work. In geometry, some words, such as point, line, and plane

are undefined terms. Although these words are not formally defined, it is important to have general

agreement about what each word means.

A point _______________________________________________________________________________.

A line _________________________________________________________________________________

In this book, lines are always straight lines.

A plane ________________________________________________________________________________

A few concepts in geometry must also be commonly understood without being defined. One such concept is

the idea that a point lies on a line or a plane.

Collinear points ________________________________________________________________________.

Coplanar points ________________________________________________________________________.

Ex. 1 Decide whether the statement is true or false.

a. Point X lies on line m. b. X, Y, and Z are collinear.

c. Point W lies on line m. d. X, Y, and Z are coplanar.

e. Point V lies on line l. f. V, Y, and X are collinear.

g. X, Y, and V are collinear. h. X, Y, and V are coplanar.

Ex. 2 Name a point that is collinear with the given points.

a. B and E b. C and H

c. D and G d. A and C

e. H and E f. G and B

g. B and I h. B and C

Ex. 3 Name a point that is coplanar with the given points.

a. M, N, and R b. M, N, and O

c. M, T, and Q d. Q, T, and R

e. T, R, and S f. Q, S, and O

g. O, P, and M h. O ,S, and R

Another undefined concept in geometry is the idea that a point on a line is between two other points on the line.

You can use this idea to define other important terms in geometry.

The line segment or segment AB ( AB )__________________________________________

____________________________________________________________________________.

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JK

The ray AB (

AB )_____________________________________________________________

____________________________________________________________________________.

Note that ______ is the same as _______, and ______ is the same as _______, but ______ and ______ are not

the same. They have different initial points and extend in different directions.

Ex. 4 Draw three noncollinear points, J, K, and L. Then draw , , KL ,

LJ

If C is between A and B, then

CA and

CB are _________________________________.

Ex. 5 Draw two lines. Label points on the lines and name two pairs of opposite rays.

Ex. 6 Complete the sentence.

a. AB consists of the endpoints A and B and all points on the line that lie ________________________.

b.

PQ consists of the initial point P and all points on the line that lie ___________________________.

c. Two rays or segments are collinear if they ___________________________________________________.

d.

MN and

ML are opposite rays if ________________________________________________________.

GOAL 2: Sketching intersections of Lines and Planes. Ex. 6 Sketch the figure described.

a. Three points that are coplanar but not collinear. b. Three lines that intersect at a single point.

c. Three lines that intersect at two points. d. Three lines that intersect at three points.

e. Two planes that intersect. e. Two planes that do not intersect.

AB

PQ

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Section 1.3 Segments and Their Measures GOAL 1: Using Angle Postulates

A postulate or axiom is a ____________________________________________________________________.

A theorem ________________________________________________________________________________.

POSTULATE 1 Ruler Postulate

The absolute value of a number is the distance it is from 0.

Ex, 1 Evaluate the following.

│− 4│ │4│ │4 − 7│ │7 − 4│ │− 4 − 7│ │−7 + 4│ │− 4 + 7│

Ex. 2 Measure the length of the segment to the nearest millimeter. A

B

POSTUTLATE 2 Segment Addition Postulate

Ex. 3 Draw a sketch of the three collinear points. Then write the Segment Addition Postulate for the points.

a. T is between M and N. b. J is between S and H.

Ex. 4 In the diagram of collinear points, GK = 24, HJ = 10, and GH = HI = IJ.

Find each length.

a. HI b. IJ c. GH

d. JK e. IG f. IK

The points on a line can be matched one to one with the real numbers. The real numbers that

corresponds to a point is the coordinate of the point.

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

the coordinates of A and B. d = │A - B│ or │B - A│

AB is also called the length of AB .

If B is between A and C, then AB +BC = AC.

If AB + BC = AC, then B is between A and C. A B C

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Ex. 5 Suppose J is between H and K. Use the Segment Addition Postulate to solve for x. Then find

the length of each segment.

a. HJ = 2x + 4 b. HJ = 5x -- 3

JK = 3x + 3 JK = 8x -- 9

KH = 22 KH = 131

GOAL 2: Using the Distance Formula The Distance Formula is a ___________________________________________________________________

_________________________________________________________________________________________.

Ex. 6 Evaluate each expression.

a. 22 34 b. 22

52 c. 22 46

The Distance Formula

Ex. 7 Find the distance between each pair of points.

D(1, 3), E(−2, 4), F(0, −4)

DE = DF = EF =

Ex. 8 Review: Plot A(1, −4) and B(−3, 2) on the graph.

Congruent segments are ____________________________________________________________________.

There is a special symbol, , for indicating congruence.

Lengths are equal Segments are congruent

AB = AD AB AD

Ex. 9 Find the lengths of the segments. Tell whether any of the segments have the same length.

If 11, yxA and 22 , yxB are points in a coordinate plane, then the distance between A and B is

.2

12

2

12 yyxxAB

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Section 1.4 Angles and their measures

GOAL: 1 Using Angle Postulates

An angle _________________________________________________________________________________.

The rays are the ______________ of the angle. The initial point is the ________________ of the angle.

Two angles are adjacent angles if ____________________________________________________________

______________________________ The angle that has sides AB and AC is denoted by BAC , CAB , .A

The point A is the vertex of the angle.

Ex. 1 Name the vertex and sides of the angle. Write two names for the angle.

Ex. 2 Name the angles in the figure.

The measure of A is denoted by _____________. The measure of an angle can be approximated with a

protractor, using units called degrees (◦).

Congruent angles _________________________________________________________________________.

Measures are equal Angles are congruent

DEFmBACm DEFBAC

“is equal to” “is congruent to”

POSTULATE 3 Protractor Postulate

Ex. 3 Use a protractor to measure each angle to the nearest degree.

a. b.

A point is in the interior of an angle if it is _____________________ _________________________________.

A point is in the exterior of an angle if it is ______________________________________________________.

POSTULATE 4 Angle Addition Postulate

Consider a point A on one side of OB. The rays of the form OA

can be matched one to one with the real numbers 0 to 180.

The measure of AOB is equal to the absolute value of the

difference between the real numbers for OA and OB.

If P is in the interior of RST , then

RSTmPSTmRSPm .

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Ex. 4 Use the Angle Addition Postulate to find the measure of the unknown angle.

a. b. mDEF ____ mABC ____

GOAL 2: Classifying Angles Angles are classified as acute, right, obtuse, and straight, according to their measures. Angles have measures

greater than 0° and less than or equal to 180°.

Ex. 5 State whether the angle appears to be acute, right, obtuse, or straight. Then estimate its measure.

a. b. c.

Ex. 6 In a coordinate plane,

1. Plot the points and sketch .ABC

2. Classify the angle.

3. Write the coordinates of a point that lies in the interior of the angle and

4. Write the coordinates of a point that lies in the exterior of the angle.

a. A(2, −4) b. A(−2, 1)

B(−1, −1) B(1, 4)

C(4, 1) C(7, 2)

Section 1.5 Segment and Angle Bisectors

GOAL 1: Bisecting a Segment The midpoint of a segment _________________________________________________________________.

In this book, matching red congruence marks identify congruent segments in diagrams.

A segment bisector is a ____________________________________________________________________.

Ex. 1 Use a compass and a straightedge (ruler) to construct a segment bisector and midpoint of AB. (Page 34)

This will be turned in today during class. Explain here how to construct a segment bisector.

If you know the coordinates of the endpoints of a segment, you can calculate the coordinates of the midpoint.

You simply take the mean, or average of the x-coordinates and the average of the y-coordinates. (Add them and

take half.) This is known as the midpoint formula.

2,

2

2121 yyxx

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Ex. 2 Find the coordinates of the midpoint of a segment with the given endpoints.

a. A(−2, 4) b. C(2, 4) c. E(−3, 2)

B(4, 6) D(0, –8) F(7, −5)

Ex. 3 Find the coordinateds of the other endpoint of the segment with the given endpoint and the midpoint M.

a. A(− 8, −1) b. B( 3, 5)

M( 0, 3) M( 7, −4)

GOAL 2: Bisecting an Angle

An angle bisector is a ____________________________________________________

______________________________________________________________________.

Ex 4 Draw an angle with a straight edge. Use construction tools to find the bisector of the angle (page 36).

This will be turned in today. Explain here how you bisect an angle bisector.

Ex. 5 PTu ruu

is the angle bisector of RPS . Find the two angle measures not given in the diagram.

a. b. c.

Ex. 6 BTu ruu

bisects .ABC Find the value of x.

a. b.

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Section 1.6 Angle Pair Relationships

GOAL 1: Vertical and Angles and Linear Pairs

Two angles are vertical angles if _____________________________________________________________.

Two adjacent angles are a linear pair if ________________________________________________________.

Ex. 1 Use the figure at the right.

a. Are 1 and 2 adjacent?

b. Are 1 and 2 a linear pair?

c. Are 3 and 4 a linear pair?

d. Are 2 and 5 vertical angles?

e. Are 1 and 4 vertical angles?

f. Are 3 and 5 vertical angles?

Two important facts are listed below. We will study them more in Chapter 2.

Vertical angles are congruent.

The sum of the measures of angles that form a linear pair is 180°.

Ex. 2 Decide whether the statement is always, sometimes, or never true.

a. If m 1 = 40°, then m 3 = 140°

b. If m 4 = 130°, then m 3 = 50°.

c. 1 and 3 are congruent.

d. m 1 + m 3 = m 2 + m 4.

e. m 2 = 180m3 .

Ex. 3 Use the figure at the right.

a. If m 6 = 78°, then m 7 = _______.

b. If m 8 = 94°, then m 6 = _______.

c. If m 9 = 124, then m 8 = _______.

d. If m 7 = 47°, then m 9 = _______.

e. If m 8 = 158°, then m 9 = _______.

f. If m 7 = 15°, then m 6 = _______.

Ex. 4 Find the value of the variable.

a.

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GOAL 2: Complementary and Supplementary Angles

Two angles are complementary angles if _____________________________________________________.

Each angle is the complement of the other. Complementary angles can be adjacent or nonadjacent.

Two angles are supplementary angles if ______________________________________________________.

Each angle is a supplement of the other. Supplementary angles can be adjacent or nonadjacent.

Ex. 5 State whether the angles are complementary, supplementary, or¸ neither.

a. b. c.

Ex. 6 Assume A and B are complementary and B and C are supplementary.

a. If mA = 42°, then mB = ________ and mC = ________.

b. If mB = 78°, then mA = ________ and mC = ________.

c. If mA = 17°, then mB = ________ and mC = ________.

d. If mB = 45°, then mA = ________ and mC = ________.

Ex. 7 A and B are complementary. Find the m A and the mB.

a. b.

85 xAm 73 xAm

4 xBm 111 xBm

Ex. 8 A and B are supplementary. Find the mA and the mB.

a. b.

xAm 3 16 xAm

8 xBm 175 xBm

Ex. 9 Find the values of the variables.

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Section 1.7

GOAL 1: Reviewing Perimeter, Circumference, and Area In this lesson, you will review some common formulas for perimeter, circumference, and area. You will learn

more about area in Chapters 6, 11, and 12.

Ex. 1 Find the perimeter (or circumference) and area of the figure.

Ex. 2 Find the area of the figure described.

a. Rectangle with length 8 centimeters and width 4.5 centimeters.

b. Triangle with height 5 inches and base 12 inches.

c. Circle with diameter 10 feet.