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Geometry 1 Unit 1: Basics of Geometry

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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.

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 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.

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

- Conjecture
- An unproven statement that is based on observations.

- Inductive Reasoning
- The process of looking for patterns and making conjectures.

EXAMPLE 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

Conjecture: Five students can shake hands in

6 + 4, or 10 different ways.

EXAMPLE 3

Make a conjecture

= 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

= 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.

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

- 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.

- An example that shows a conjecture is false.

EXAMPLE 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

ANSWER

Because a counterexample exists, the conjecture is false.

EXAMPLE 5

Find a counterexample

–2 +–3

=–5

5.

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

1.2: Points, Lines and Planes

- 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.

- 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.

Point

- The most basic building block of Geometry
- Has no size
- A location in space
- Represented with a dot
- Named with a Capital Letter

Example: point P

P

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

Example: line AB, AB, BA or l

B

A

**2 points determine a line

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

Example: Plane P or plane ABC

AC

B P

**3 noncollinear points determine a plane

C

B

A

- Collinear
- Points that lie on the same line
Points A, B, and C are Collinear

- Points that lie on the same line

E

F

D

- Coplanar
- Points that lie on the same plane
Points D, E, and F are Coplanar

- Points that lie on the same plane

- 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 BA B

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

- 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

D

A

B

F

C

- 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

C

B

- Opposite Rays
- If C is between A and B, then CA and CB are opposite rays.
- Together they make a line.

C YD C Y DCY 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

- The intersection of two lines is a point.
- The intersection of two planes is a line.

Unit 1-Basics of Geometry

1.3: Segments and Their Measures

- Postulates
- Rules that are accepted without proof.
- Also called axioms

- 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.

- 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

- Example 1
- Measure the length of the segment to the nearest millimeter.

D

E

E

D

A

B

F

C

- 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

A

B

C

AB

BC

- 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.

- 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?

- 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

- Example 3
- Find the lengths of the segments. Tell whether any of the segments have the same length.

- 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.

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

A2.5B

AB = DC = 2.5

AB DC

D2.5CAD BC

B(x2, y2)

c

|y2 – y1|

a

A(x1, y1)

C(x2, y1)

b

|x2 – x1|

- Distance Formula and Pythagorean Theorem

(AB)2 = (x2 – x1)2 + (y2 – y1)2

c2 = a2 + b2

- 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?

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

Unit 1-Basics of Geometry

1.4: Angles and Their Measures

- 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

A

T

- When naming an angle, the vertex must be the middle letter.
angle CAT, angle TAC, CAT or TAC

C

A

T

- If a vertex has only one angle then you can name it with that letter alone.
TAC could also be called A.

D

A

1

C

B

- Example 1
- Name all the angles in the following drawing

- 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.

- Measure of an Angle
- The smallest rotation between the two sides of the angle.

- Congruent angles
- Angles that have the same measure.

- Angle measure notation
- Use an m before the angle symbol to show the measure:
mA = 34º or measure of A = 34º

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

A

O

B

- 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.

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.

- 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

- Angle Addition Postulate
- If P is in the interior of RST, then
mRSP + mPST = mRST

- If P is in the interior of RST, then

R

m RST

m RSP

S

P

m PST

T

Left only

Right only

Both bulbs

- 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?

- Acute Angle
- An angle whose measure is greater than 0° and less than 90º.

- Right Angle
- An angle whose measure is 90º

- Obtuse Angle
- An angle whose measure is greater than 90º and less than 180º.

- Straight Angle
- An angle whose measure is 180°.

A

- 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

- Plot the following points.

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

C

A

B

D

- 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

1.5: Segment and Angle Bisectors

- 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).

M is the midpoint of LN

L M N

LM MN

- Segment bisector
- A segment, ray, line, or plane that intersects a segment at its midpoint.

- Compass
- Geometric tool that is used to construct circles and arcs.

- Straightedge
- Ruler without marks.

- Construction
- Geometric drawing that uses a compass and straightedge.

- 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.

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

- Midpoint Formula
- Given two points (x1, y1) and (x2, y2) the coordinates of the midpoint are:
x1 + x2,y1 + y2

2 2

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

- Example 1
- Find the coordinates of the midpoint of the segment with endpoints at (12, -8) and (-3, 15).

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

- Example 3
- One endpoint is (17,-3) and the midpoint is (8,2).
Find the coordinates of the other endpoint.

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

- Example 4
- One endpoint is (-5,8) and the midpoint is (6,3). Find the coordinates of the other endpoint.

A

D

C

mACD = mBCD

B

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

- 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.

- Example 5
- JK bisects HJL. Given that mHJL = 42°, what are the measures of HJK and KJL?

47°

wire

wire

Cellular phone tower

- 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.

- 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

1.6 Angle Pair Relationships

- Vertical Angles
- Angles whose sides form opposite rays.

1 and 3 are vertical angles.

2 and 4 are vertical angles.

1

4

2

3

- 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

3

1

4

5

- 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?

- Example 2

M

(4x + 15)°

L

(5x + 30)°

P

N

(3y + 15)°

(3y – 15)°

O

- Example 3
- Solve for x and y. Then find the angle measures.

- Complementary Angles
- Two angles that have a sum of 90º
- Each angle is a complement of the other.

- Supplementary Angles
- Two angles that have a sum of 180º
- Each angle is a supplement of the other.

- 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.

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

- Example 5
- Given that G is a supplement of H and mG is 82°, find mH.
- Given that U is a complement of V, and mU is 73°, find mV.

- Example 6
- T and S are supplementary.
The measure of T is half the measure of S. Find mS.

- T and S are supplementary.

- Example 7
- D and E are complements and D and F are supplements. If mE is four times mD, find the measure of each of the three angles.

Unit 1-Basics of Geometry

1.7: Introduction to Perimeter, Circumference, and Area

- Square
- Side length s
- P = 4s
- A = s2

s

- Rectangle
- Length land width w
- P = 2l + 2w
- A = lw

l

w

- Triangle
- Side lengths a, b, and c,
- Base b, and height h
- P = a + b + c
- A = ½bh

a

c

h

b

- 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

- Example 1
- Find the perimeter and area of a rectangle of length 4.5m and width 0.5m.

- 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.

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

- 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?

- 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?

- 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?