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# Geometry 1 Unit 1: Basics of Geometry - PowerPoint PPT Presentation

Geometry 1 Unit 1: Basics of Geometry. Geometry 1 Unit 1. 1.1 Patterns and Inductive Reasoning. Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure. Each circle is divided into twice as many equal regions as the figure number.

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### Geometry 1 Unit 1

1.1 Patterns and Inductive Reasoning

Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure.

Each circle is divided into twice as many equal regions as the figure number.

Sketch the fourth figure by dividing a circle into eighths.

Shade the section just above the horizontal segment at the left.

EXAMPLE 1

Describe a visual pattern

SOLUTION

Sketch the fifth figure in the pattern in example Then sketch the fourth figure. 1.

for Examples 1 and 2

GUIDED PRACTICE

Notice that each number in the pattern is three times the previous number.

Continue the pattern. The next three numbers are –567, –1701, and –5103.

EXAMPLE 2

Describe a number pattern

Describe the pattern in the numbers –7, –21, –63, –189,… and write the next three numbers in the pattern.

2 previous number..

Describe the pattern in the numbers 5.01, 5.03, 5.05, 5.07,… Write the next three numbers in the pattern.

Notice that each number in the pattern is increasing by 0.02.

5.01

5.03

5.05

5.09

5.11

5.13

5.07

+0.02

+0.02

+0.02

+0.02

+0.02

+0.02

Continue the pattern. The next three numbers are 5.09, 5.11 and 5.13

for Examples 1 and 2

GUIDED PRACTICE

Patterns and Inductive Reasoning previous number.

• Conjecture

• An unproven statement that is based on observations.

• Inductive Reasoning

• The process of looking for patterns and making conjectures.

EXAMPLE previous number.3

Make a conjecture

Given five students, make a conjecture about the number of different handshakes that can take place.

SOLUTION

Make a table and look for a pattern. Notice the pattern in how the number of connections increases. You can use the pattern to make a conjecture.

Conjecture: Five students can shake hands in

6 + 4, or 10 different ways.

EXAMPLE 3

Make a conjecture

= previous number.4 3

= 8 3

= 113

= 17 3

Conjecture: The sum of any three consecutive integers is three times the second number.

EXAMPLE 4

Make and test a conjecture

Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive integers.

SOLUTION

STEP 1

Find a pattern using a few groups of small numbers.

3 + 4 + 5

= 12

7 + 8 + 9

= 12

10 + 11+ 12

= 33

16 + 17 + 18

= 51

= previous number.101 3

= 0 3

EXAMPLE 4

Make and test a conjecture

STEP 1

Test your conjecture using other numbers. For example, test that it works with the groups –1, 0, 1 and 100, 101, 102.

100 + 101 + 102

= 303

–1 + 0 + 1

= 0

3 previous number..

Make and test a conjecture about the sign of the product of any three negative integers.

Conjecture: The result of the product of three negative numbers is a negative number.

Test:

Test conjecture using the negative integers –2, –5 and –4

–2 –5 –4

= –40

for Examples 3 and 4

GUIDED PRACTICE

Patterns and Inductive Reasoning previous number.

• Counterexample

• An example that shows a conjecture is false.

All Math teachers are male.

Mrs. Beery, Ms. Wildermuth, Mrs. Hodge, Mrs. Cherry, Mrs. Frimer, Mrs. Dolezal are all counterexamples.

EXAMPLE previous number.5

Find a counterexample

A student makes the following conjecture about the sum of two numbers. Find a counterexample to disprove the student’s conjecture.

Conjecture: The sum of two numbers is always greater than the larger number.

SOLUTION

To find a counterexample, you need to find a sum that is less than the larger number.

–5 > –2 previous number.

Because a counterexample exists, the conjecture is false.

EXAMPLE 5

Find a counterexample

–2 +–3

=–5

5. previous number.

Find a counterexample to show that the following conjecture is false.

=

12

14

14

12

( )2

>

Because a counterexample exist, the conjecture is false

for Examples 5 and 6

GUIDED PRACTICE

Conjecture: The value of x2 is always greater than the value of x.

### Unit 1-Basics of Geometry previous number.

1.2: Points, Lines and Planes

Points, Lines, and Planes previous number.

• Definition

• Uses known words to describe a new word.

• Undefined terms

• Words that lack a formal definition.

• In Geometry it is important to have a general agreement about these words.

• The building blocks of Geometry are undefined terms.

Points, Lines, and Planes previous number.

• The 3 Building Blocks of Geometry:

• Point

• Line

• Plane

• These are called the “building blocks of geometry” because these terms lay the foundation for Geometry.

Points, Lines, and Planes previous number.

Point

• The most basic building block of Geometry

• Has no size

• A location in space

• Represented with a dot

• Named with a Capital Letter

Points, Lines, and Planes previous number.

Example: point P

P

Points, Lines, and Planes previous number.

Line

• Set of infinitely many points

• One dimensional, has no thickness

• Goes on forever in both directions

• Named using any two points on the line with the line symbol over them, or a lowercase script letter

l previous number.

Points, Lines, and Planes

Example: line AB, AB, BA or l

B

A

**2 points determine a line

Points, Lines, and Planes previous number.

Plane

• Has length and width, but no thickness

• A flat surface that extends infinitely in 2-dimensions (length and width)

• Represented with a four-sided figure like a tilted piece of paper, drawn in perspective

• Named with a script capital letter or 3 points in the plane

Points, Lines, and Planes previous number.

Example: Plane P or plane ABC

A C

B P

**3 noncollinear points determine a plane

C previous number.

B

A

Points, Lines, and Planes

• Collinear

• Points that lie on the same line

Points A, B, and C are Collinear

E previous number.

F

D

Points, Lines, and Planes

• Coplanar

• Points that lie on the same plane

Points D, E, and F are Coplanar

Points, Lines, and Planes previous number.

• Line Segment

• Two points (called the endpoints) and all the points between them that are collinear with those two points

Named line segment AB, AB, or BA

line AB segment AB

A B A B

Points, Lines, and Planes previous number.

• Ray

• Part of a line that starts at a point and extends infinitely in one direction.

• Initial Point

• Starting point for a ray.

• Ray CD, or CD, is part of CD that contains point C and all points on line CD that are on the same side as of C as D

• “It begins at C and goes through D and on forever”

E previous number.

D

A

B

F

C

Segments and Their Measures

• Between

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

Point B is between A and C

Point E is NOT between D and F

A previous number.

C

B

Points, Lines, and Planes

• Opposite Rays

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

• Together they make a line.

Points, Lines, and Planes previous number.

C Y D C Y D C Y D

Line CD Ray DC Ray CD

CD and CY represent the same ray.

Notice CD is not the same as DC.

ray CD is not opposite to ray DC

Points, Lines, and Planes previous number.

• The intersection of two lines is a point.

• The intersection of two planes is a line.

### Unit 1-Basics of Geometry previous number.

1.3: Segments and Their Measures

Segments and Their Measures previous number.

• Postulates

• Rules that are accepted without proof.

• Also called axioms

Segments and Their Measures previous number.

• Ruler Postulate

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

• The real number that corresponds to a point is called 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.

• AB is also called the length of AB.

Segments and Their Measures previous number.

• Segment length can be given in several different ways. The following all mean the same thing.

• A to B equals 2 inches

• AB = 2 in.

• mAB = 2 inches

Segments and Their Measures previous number.

• Example 1

• Measure the length of the segment to the nearest millimeter.

D

E

E previous number.

D

A

B

F

C

Segments and Their Measures

• Between

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

Point B is between A and C

Point E is NOT between D and F

AC previous number.

A

B

C

AB

BC

Segments and Their Measures

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

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

Segments and Their Measures previous number.

• Example 2

• Two friends leave their homes and walk in a straight line toward the others home. When they meet, one has walked 425 yards and the other has walked 267 yards. How far apart are their homes?

Segments and Their Measures previous number.

• The Distance Formula

• A formula for computing the distance between two points in a coordinate plane.

• If A(x1,y1) and B(x2,y2) are points in a coordinate plane, then the distance between A and B is

Segments and Their Measures previous number.

• Example 3

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

Segments and Their Measures previous number.

• Congruent

• Two segments are congruent if and only if they have the same measure.

• The symbol for congruence is .

• We use = between equal numbers and  between congruent figures.

Segments and Their Measures previous number.

Markings on figures are used to show congruence. Use identical markings for each pair of congruent parts.

A 2.5 B

AB = DC = 2.5

AB  DC

D 2.5 C AD  BC

B(x previous number.2, y2)

c

|y2 – y1|

a

A(x1, y1)

C(x2, y1)

b

|x2 – x1|

Segments and Their Measures

• Distance Formula and Pythagorean Theorem

(AB)2 = (x2 – x1)2 + (y2 – y1)2

c2 = a2 + b2

Segments and Their Measures previous number.

• Example 4

• On the map, the city blocks are 410 feet apart east-west and 370 feet apart north south.

• Find the walking distance between C and D.

• What would the distance be if a diagonal street existed between the two points?

### Unit 1-Basics of Geometry previous number.

1.4: Angles and Their Measures

Angles and Their Measures previous number.

• Angle

• Formed by two rays that share a common endpoint.

• Sides

• The rays that make the angle.

• Vertex

• The initial point of the rays.

C previous number.

A

T

Angles and Their Measures

• When naming an angle, the vertex must be the middle letter.

angle CAT, angle TAC, CAT or TAC

C previous number.

A

T

Angles and Their Measures

• If a vertex has only one angle then you can name it with that letter alone.

TAC could also be called A.

D previous number.

A

1

C

B

Angles and Their Measures

• Example 1

• Name all the angles in the following drawing

Angles and Their Measures previous number.

• Protractor

• Geometry tool used to measure angles. Angles are measured in Degrees.

• Things to know

• A full circle is 360 degrees, or 360º.

• A line is 180º.

Angles and Their Measures previous number.

• Measure of an Angle

• The smallest rotation between the two sides of the angle.

• Congruent angles

• Angles that have the same measure.

Angles and Their Measures previous number.

• Angle measure notation

• Use an m before the angle symbol to show the measure:

mA = 34º or measure of A = 34º

A previous number.

O

B

Angles and Their Measures

• Protractor Postulate

• Consider a point A not on OB. The rays of the form OA can be matched one to one with the real numbers from 0 to 180.

• The measure of an angle is equal to the number on the protractor which one side of the angle passes through when the other side goes through the number zero on the same scale.

Angles and Their Measures previous number.

Step 1: Place the center mark of the protractor on the vertex.

Step 2: Line up the 0-mark with one side of the angle.

Step 3: Read the measure on the protractor scale.

**Be sure you are reading the scale with the 0-mark you are using.

Angles and Their Measures previous number.

• Interior

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

• Exterior

• A point is in the exterior of an angle if it is not on the angle or in its interior.

E

D

exterior

interior

Angles and Their Measures previous number.

• If P is in the interior of RST, then

mRSP + mPST = mRST

R

m RST

m RSP

S

P

m PST

T

Left only previous number.

Right only

Both bulbs

Angles and Their Measures

• Example 2

• The backyard of a house is illuminated by a light fixture that has two bulbs.

• Each bulb illuminates an angle of 120°.

• If the angle illuminated only by the right bulb is 35°, what is the angle illuminated by both bulbs?

Angles and Their Measures previous number.

• Acute Angle

• An angle whose measure is greater than 0° and less than 90º.

Angles and Their Measures previous number.

• Right Angle

• An angle whose measure is 90º

Angles and Their Measures previous number.

• Obtuse Angle

• An angle whose measure is greater than 90º and less than 180º.

Angles and Their Measures previous number.

• Straight Angle

• An angle whose measure is 180°.

A

Angles and Their Measures previous number.

• Example 3

• Plot the following points.

• A(-3, -1), B(-1, 1), C(2, 4), D(2, 1), and E(2, -2)

• Measure and classify the following angles as acute, right, obtuse or straight.

a. DBE

b. EBC

c. ABC

d. ABD

Angles and Their Measures previous number.

Angles and Their Measures previous number.

• Angles that share a common vertex and side, but have no common interior points.

C

A

B

D

Angles and Their Measures previous number.

• Example 4

• Use a protractor to draw two adjacent angles LMN and NMO so that LMN is acute and LMO is straight.

### Unit 1-Basics of Geometry previous number.

1.5: Segment and Angle Bisectors

Segment and Angle Bisectors previous number.

• Midpoint

• The point on the segment that is the same distance from both endpoints.

• This point bisects the segment.

• Bisect

• To cut in half (two equal pieces).

Segment and Angle Bisectors previous number.

M is the midpoint of LN

L M N

LM  MN

Segment and Angle Bisectors previous number.

• Segment bisector

• A segment, ray, line, or plane that intersects a segment at its midpoint.

Segment and Angle Bisectors previous number.

• Compass

• Geometric tool that is used to construct circles and arcs.

• Straightedge

• Ruler without marks.

• Construction

• Geometric drawing that uses a compass and straightedge.

Segment and Angle Bisectors previous number.

• Construct a Segment Bisector and Midpoint

• Use the following steps to construct a bisector of AB and find the midpoint M of AB.

• Place the compass point at A. Use a compass setting greater than half of AB. Draw an arc.

• Keep the same compass setting. Place the compass point at B. Draw an arc. It should intersect the other arc in two places.

• Use a straightedge to draw a segment through the points of intersection. This segment bisects AB at M, the midpoint of AB.

Segment and Angle Bisectors previous number.

• Midpoint Formula

• Given two points (x1, y1) and (x2, y2) the coordinates of the midpoint are:

x1 + x2 , y1 + y2

2 2

Segment and Angle Bisectors previous number.

• Example 1

• Find the coordinates of the midpoint of the segment with endpoints at (12, -8) and (-3, 15).

Segment and Angle Bisectors previous number.

• Example 2

• Find the coordinates of the midpoint of the segment with endpoints at (5, 8) and (7, -2).

Segment and Angle Bisectors previous number.

• Example 3

• One endpoint is (17,-3) and the midpoint is (8,2).

Find the coordinates of the other endpoint.

Segment and Angle Bisectors previous number.

• Example 4

• One endpoint is (-5,8) and the midpoint is (6,3). Find the coordinates of the other endpoint.

A previous number.

D

C

mACD = mBCD

B

Segment and Angle Bisectors

• Angle bisector

• A ray that divides an angle into two adjacent angles that are congruent.

Segment and Angle Bisectors previous number.

• Construct an Angle Bisector

• Place the compass point at C. Draw an arc that intersects both sides of the angle. Label the intersections A and B.

• Place the compass point at A. Draw another arc. Then place the compass point at B. Using the same compass setting, draw a third arc to intersect the second one.

• Label the intersection D. Use a straightedge to draw a ray from C through D. This is the angle bisector.

Segment and Angle Bisectors previous number.

• Example 5

• JK bisects HJL. Given that mHJL = 42°, what are the measures of HJK and KJL?

47° previous number.

wire

wire

Cellular phone tower

Segment and Angle Bisectors

• Example 6

• A cellular phone tower bisects the angle formed by the two wires that support it. Find the measure of the angle formed by the two wires.

Segment and Angle Bisectors previous number.

• Example 7

• MO bisects LMN. The measures of the two congruent angles are (3x – 20)° and (x + 10) °. Solve for x.

### Unit 1-Basics of Geometry previous number.

1.6 Angle Pair Relationships

Angle Pair Relationships previous number.

• Vertical Angles

• Angles whose sides form opposite rays.

1 and 3 are vertical angles.

2 and 4 are vertical angles.

1

4

2

3

Angle Pair Relationships previous number.

• Linear Pair of Angles

Angles that share a common vertex and a common side. Their non-common sides form a line.

5 and 6 are a linear pair of angles.

5

6

2 previous number.

3

1

4

5

Angle Pair Relationships

• Example 1

• Are 1 and 2 a linear pair?

• Are 4 and 5 a linear pair?

• Are 5 and 3 vertical angles?

• Are 1 and 3 vertical angles?

Angle Pair Relationships previous number.

• Example 2

M previous number.

(4x + 15)°

L

(5x + 30)°

P

N

(3y + 15)°

(3y – 15)°

O

Angle Pair Relationships

• Example 3

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

Angle Pair Relationships previous number.

• Complementary Angles

• Two angles that have a sum of 90º

• Each angle is a complement of the other.

Angle Pair Relationships previous number.

• Supplementary Angles

• Two angles that have a sum of 180º

• Each angle is a supplement of the other.

Angle Pair Relationships previous number.

• Example 4

• State whether the two angles are complementary, supplementary or neither.

• The angles formed by the hands of a clock at 11:00 and 1:00.

Angle Pair Relationships previous number.

• Example 5

• Given that G is a supplement of H and mG is 82°, find mH.

• Given that U is a complement of V, and mU is 73°, find mV.

Angle Pair Relationships previous number.

• Example 6

• T and S are supplementary.

The measure of T is half the measure of S. Find mS.

Angle Pair Relationships previous number.

• Example 7

• D and E are complements and D and F are supplements. If mE is four times mD, find the measure of each of the three angles.

### Unit 1-Basics of Geometry previous number.

1.7: Introduction to Perimeter, Circumference, and Area

Introduction to Perimeter, Circumference, and Area previous number.

• Square

• Side length s

• P = 4s

• A = s2

s

Introduction to Perimeter, Circumference, and Area previous number.

• Rectangle

• Length land width w

• P = 2l + 2w

• A = lw

l

w

Introduction to Perimeter, Circumference, and Area previous number.

• Triangle

• Side lengths a, b, and c,

• Base b, and height h

• P = a + b + c

• A = ½bh

a

c

h

b

Introduction to Perimeter, Circumference, and Area previous number.

• Circle

• C = 2π r

• A = π r2

• Pi (π) is the ratio of the circle’s circumference to its diameter. π ≈ 3.14

r

Introduction to Perimeter, Circumference, and Area previous number.

• Example 1

• Find the perimeter and area of a rectangle of length 4.5m and width 0.5m.

Introduction to Perimeter, Circumference, and Area previous number.

• Example 2

• A road sign consists of a pole with a circular sign on top. The top of the circle is 10 feet high and the bottom of the circle is 8 feet high.

• Find the diameter, radius, circumference and area of the circle. Use π ≈ 3.14.

Introduction to Perimeter, Circumference, and Area previous number.

• Example 3

• Find the area and perimeter of the triangle defined by H(-2, 2), J(3, -1), and K(-2, -4).

Introduction to Perimeter, Circumference, and Area previous number.

• Example 4

• A maintenance worker needs to fertilize a 9-hole golf course. The entire course covers a rectangular area that is approximately 1800 feet by 2700 feet. Each bag of fertilizer covers 20,000 square feet. How many bags will the worker need?

Introduction to Perimeter, Circumference, and Area previous number.

• Example 5

• You are designing a mat for a picture. The picture is 8 inches wide and 10 inches tall. The mat is to be 2 inches wide. What is the area of the mat?

Introduction to Perimeter, Circumference, and Area previous number.

• Example 6

• You are making a triangular window. The height of the window is 18 inches and the area should be 297 square inches. What should the base of the window be?