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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 basics of geometry

Geometry 1 Unit 1: Basics of Geometry


Geometry 1 unit 1

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.

ANSWER

for Examples 1 and 2

GUIDED PRACTICE


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

ANSWER

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

ANSWER

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
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.


ANSWER previous number.

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

ANSWER

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.

ANSWER

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 reasoning1
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.

ANSWER

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

>

ANSWER

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

Unit 1-Basics of Geometry previous number.

1.2: Points, Lines and Planes


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 planes1
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 planes2
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 planes3
Points, Lines, and Planes previous number.

Example: point P

P


Points lines and planes4
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


Points lines and planes5

l previous number.

Points, Lines, and Planes

Example: line AB, AB, BA or l

B

A

**2 points determine a line


Points lines and planes6
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 planes7
Points, Lines, and Planes previous number.

Example: Plane P or plane ABC

A C

B P

**3 noncollinear points determine a plane


Points lines and planes8

C previous number.

B

A

Points, Lines, and Planes

  • Collinear

    • Points that lie on the same line

      Points A, B, and C are Collinear


Points lines and planes9

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 planes10
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 planes11
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”


Segments and their measures

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


Points lines and planes12

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 planes13
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 planes14
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 geometry1

Unit 1-Basics of Geometry previous number.

1.3: Segments and Their Measures


Segments and their measures1
Segments and Their Measures previous number.

  • Postulates

    • Rules that are accepted without proof.

    • Also called axioms


Segments and their measures2
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 measures3
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 measures4
Segments and Their Measures previous number.

  • Example 1

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

D

E


Segments and their measures5

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


Segments and their measures6

AC previous number.

A

B

C

AB

BC

Segments and Their Measures

  • Segment Addition Postulate

    • 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 measures7
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 measures8
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 measures9
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 measures10
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 measures11
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


Segments and their measures12

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 measures13
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 geometry2

Unit 1-Basics of Geometry previous number.

1.4: Angles and Their Measures


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.


Angles and their measures1

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


Angles and their measures2

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.


Angles and their measures3

D previous number.

A

1

C

B

Angles and Their Measures

  • Example 1

    • Name all the angles in the following drawing


Angles and their measures4
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 measures5
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 measures6
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


Angles and their measures7

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 measures8
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 measures9
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 measures10
Angles and Their Measures previous number.

  • Angle Addition Postulate

    • 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


Angles and their measures11

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 measures12
Angles and Their Measures previous number.

  • Acute Angle

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


Angles and their measures13
Angles and Their Measures previous number.

  • Right Angle

    • An angle whose measure is 90º


Angles and their measures14
Angles and Their Measures previous number.

  • Obtuse Angle

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


Angles and their measures15
Angles and Their Measures previous number.

  • Straight Angle

    • An angle whose measure is 180°.

A


Angles and their measures16
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 measures17
Angles and Their Measures previous number.


Angles and their measures18
Angles and Their Measures previous number.

  • Adjacent Angles

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

C

A

B

D


Angles and their measures19
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 geometry3

Unit 1-Basics of Geometry previous number.

1.5: Segment and Angle Bisectors


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 bisectors1
Segment and Angle Bisectors previous number.

M is the midpoint of LN

L M N

LM  MN


Segment and angle bisectors2
Segment and Angle Bisectors previous number.

  • Segment bisector

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


Segment and angle bisectors3
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 bisectors4
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 bisectors5
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 bisectors6
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 bisectors7
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 bisectors8
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 bisectors9
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.


Segment and angle bisectors10

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 bisectors11
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 bisectors12
Segment and Angle Bisectors previous number.

  • Example 5

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


Segment and angle bisectors13

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 bisectors14
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 geometry4

Unit 1-Basics of Geometry previous number.

1.6 Angle Pair Relationships


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 relationships1
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


Angle pair relationships2

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 relationships3
Angle Pair Relationships previous number.

  • Example 2


Angle pair relationships4

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 relationships5
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 relationships6
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 relationships7
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 relationships8
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 relationships9
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 relationships10
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 geometry5

Unit 1-Basics of Geometry previous number.

1.7: Introduction to Perimeter, Circumference, and Area


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 area1
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 area2
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 area3
Introduction to Perimeter, Circumference, and Area previous number.

  • Circle

    • Radius r

    • 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 area4
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 area5
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 area6
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 area7
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 area8
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 area9
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?


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