Lesson 1.1 Patterns and Inductive Reasoning

Lesson 1.1Patterns and

Inductive ReasoningYou will learn to…* find and describe patterns* use inductive reasoning to make conjectures

Sketch the next figure.

1.

2.2 84

14

16

Describe the pattern.Find the next three numbers.

3. 0, 1, 1, 2, 3, 5, 8,…

4. 4, 9, 16, 25, …

5. 1 2 3 4 2 3 4 5

13, 21, 34

36, 49, 64

__ , __ , __ , __ , 5 6 76 7 8

__ , __ , __

How many squares are in the next object?

6.

7.

3, 6, 10,

1, 4, 9,

15

16

A conjecture is an unproven statement that is

based on a pattern.

Inductive Reasoning is the process of looking for a

pattern and making a conjecture.

Complete the conjecture.8. The sum of any 2 odd

numbers is __________.even

odd9. The product of any 2 odd numbers is __________.

A counterexample is an example that shows a

conjecture is false.

Find a counterexample.

10. The sum of 2 numbers is always greater than the larger of the numbers.

11. If a shape has 2 sides the same length, then it must be a rectangle.

Describe the pattern. Find the next numbers or letters in the sequence.

12.

13. 14. A, 2, B, 0, C, 2, D, 0, E, 3,…

J, F, M, A, …

O, T, T, F, …

F, 3, G, 2, H, 4, …

M, J, J, A, S, O , N, D

F, S, S, E, N, T, …

WorkbookPage 1 (1-5)

Lesson 1.2Points, Lines,

and PlanesYou will learn to…* understand and use the basic

geometry terms* sketch intersections of lines and planes

Undefined terms cannot be mathematically defined using

other known words.

point

lineplane

Two points determine a line.

T

Postulate 2Three points determine a plane.

A B

C

plane ABC or plane T

Do you see…plane ABF ? plane ADG?

A B

CD

E FG

H

Collinear points

Coplanar points

Coplanar lines

A BC

AB

C

points that lie on the same line

points that lie in the same plane

lines that lie in the same plane

Betweeness refers to collinear points only.

Point B is between A and C.

A B C

Line AB

AB BA

n

line n

ABC

CA

BC

Segment AB

AB BA

A

B

A

B

Is AC the same as AB?

CNO

Ray AB

AB BA

AB

NA NB

AB

N

Opposite rays form __________.

?

a line

Opposite Rays share an endpoint

1) Draw three noncolinear points J, K, and L.

2) Draw JK, KL, and LJ.

JK

L

If 2 lines intersect, then their intersection is ____________. a point

If 2 planes intersect, then their intersection is ___________. a line

If a line intersects a plane they intersect at __________.a point

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Lesson 1.3Segments & Their

MeasuresYou will learn to…* use segment postulates* use the distance formula

The distance between points A and B is written as AB which is the length of AB.

-3 -2 -1 0 1 2 3

A B

AB = | - 2 – 3| or | 3 – - 2| = 5

Distance is the absolute value of their difference.

1. Find the length of the segment.

134

inches168

=

2. Find the length of the segment.

112

inches

238

78

Draw a segment that is….

3. 4 cm long

4. 2.7 cm long

5. 56 mm long

A postulate is a statement or rule that is accepted

without proof.

Rules that are proven are called theorems.

Segment Addition PostulateIf B is between A and C,

then AB + BC = AC.

A B CAC

BCAB

6. Find AB.20

A B C50

AB = 30AB + 20 = 50

7. Write an expression for AC.

3x + 2A B C

?

AC =

2x - 5

AC = (2x – 5) + (3x + 2)

5x - 3

8. Write an expression for AC.

8x + 1A C E

12x + 10

AC = 4x + 9AC + (8x + 1) = 12x + 10

9. Suppose M is between L and N. Use the Segment Addition Postulate to solve for x.

5x + 3 = 233x + 8 + 2x – 5 = 23

LM = 3x + 8MN = 2x – 5LN = 23

5x = 20x = 4

ML N

10. E is between H and R. A is between E and R. R is between E and T. HT = 50, ER = 20, and HE=EA=AR. Find RT.

EA = AR =

H E A R T

50

20

10

1010

HE = 10

10

RT = 20

11. Find AB and CD.

A

B

C D AB=

CD=

5

5

Congruent Segments have the same lengths.

AB = CD

AB CDlengths are equal

segments are congruent

12. Use the Pythagorean Theorem to find the distance between the points.

(1,2)

(4,6)

3

4c2= 32 + 42

c2= 9 + 16

c2= 25c= 5

(4-1)2 + (6-2)2?

(x2-x1)2 + (y2-y1)2

Distance Formula

If A (x1, y1) and B (x2, y2) are points in a coordinate plane, then

AB = (x2 – x1)2 + (y2 – y1)2

13. Find TM T (3, -2) and M (1, 4)

TM = (1 - 3)2 + (4 - -2)2

TM = (-2)2 + (6)2

TM = 4 + 36 = 40 ≈ 6.324 · 10 = 2 10

Tiled Floor

Street Corner

Two-Point Perspective

Street Corner

Two-Point Perspective


Lesson 1.4Angles and Their

MeasuresYou will learn to…* use angle postulates* classify angles as acute, right, obtuse, or straight

An angle consists of 2 rays that have the same endpoint

called the vertex of the angle.

YX and YZ form XYZ

X

Y

Z

60°

A

B

C

Sides:

Vertex:

BA and BC Name:

B

1. Name the sides, vertex, and angle.

54°

ABC

CBA

B

equal measures congruent angles

mABC = mXYZ

mABC mXYZ

Congruent Angles have the same measure.

Classifying Angles By Their Measures…

Acute Angle

Measure is between 0˚ and 90˚


Obtuse Angle

Measure is between 90˚ and 180˚


Right Angle

Measure is 90˚


Straight Angle

Measure is 180˚

Two angles are Adjacent Angles if they share a vertex and a side

but have no common interior points.

A

B

C

D

ABC and CBD are

adjacent angles

(not overlapping)

2. List the 3 angles shown.Which angles are adjacent?

R

S

P

T

RSP PST RST

RSP PST

Angle Addition Postulate

R

S

P

T

m RSP m PST m RST

3. Find m RST.

R

S 59º

T

P35º

94°

4. Find m RSP.

R

S 50º

T

P110º

60°

How do you measure an angle with a protractor?

5. Use a protractor to draw a 65˚ angle.

6. Use a protractor to draw a 112˚ angle.

45° or 55°?

145° or 135°?

45° or 55°?

145° or 135°?

35° or 25°?

155° or 165°?

35° or 25°?

155° or 165°?

mNPR =120° - 90°

90° - 60°30°

A

HW: 29-34

B C

FE

D

http://www.allposters.com/GetPoster.asp?CID=CC953A46B91D42FFBCFFB40485C32AAC&APNum=374827&f=c&search=9174&P=2&PP=2

http://eluzions.com/Pictures/Illusions/Ouchi.html

http://eluzions.com/Pictures/Illusions/checkerdots.html

Artist: Julian Beever

People are actually avoiding walking in the "hole"

'Make Poverty History' drawing from the side (40 ft long)


Lesson 1.5Segment and Angle

BisectorsYou will learn to…* bisect a segment* bisect an angle

A midpoint is a point that divides a segment into two

congruent segments.

midpoint

A B

M

AM = MB

AM MB

To bisect a segment means to divide it into two congruent segments.

Use a compass to locate the midpoint of a segment.

A segment bisector intersects a segment at its midpoint.

How would you find the “middle” between

2 numbers?

How would you find the “middle” between

the points?

Midpoint Formula

If A (x1, y1) and B (x2, y2) are

points in a coordinate plane,

2 2x x y y ,1 2 1 2

1. Find the midpoint between (-2, 3) and (4, -6)

(-2,3)

(4,-6)

-2 + 4 , 3 + -62 2

(1, -1.5)

2. Find the midpoint between (2, -1) and (4, -4)

2 + 4 , -1 + -42 2

(3, -2.5)

3. One endpoint is (-3, -1). The midpoint is (3, -4). Find the other endpoint.

(-3,-1), (3,-4), (x , y) -3 + x

2

= 3 -1 + y 2 = -4

-3 + x = 6 -1 + y = -8x = 9 y = -7

31

-4 1

(9, -7)

3. One endpoint is (3, -5). The midpoint is (-2, 4). Find the other endpoint.

(3,-5), (-2,4), (x , y)

3 + x = -4 -5 + y = 8x = -7 y = 13

(-7,13) 3 + x

2= -2 -5 + y

2 = 4

An angle bisector is a ray that divides an angle into

2 congruent angles.

angle bisector

4. NQ is an angle bisector. Find m MNQ.

114°

QM

PN

mMNQ = 57°

5. NQ is an angle bisector. Find m MNQ and m MNP.

64°

QM

PN

mMNQ =

mMNP =

64°

128°

6. NQ is an angle bisector. Find x.

Q

(2x – 5)°

M

PN

55°x = 30

2x – 5 = 55

7. BD bisects ABC. m ABD = (2x + 50)˚ m DBC = (5x + 5)˚ Find the measure of all 3 angles.

DA

CB

80°

x = 15 80°

mABC =

5x+5 = 2x+50

3x = 45

160°


Lesson 1.6Angle Pair

RelationshipsYou will learn to…* identify vertical angles and linear pairs* identify complementary and supplementary angles

2

A 90˚ angle forms a corner.

Two angles are Complementary Angles

if the sum of their measures is 90

A 180˚ angle forms a straight angle.

Two angles are Supplementary Angles

if the sum of their measures is 180

m A = 25°

1. Find its complement.

2. Find its supplement.

155°

65°

10°

80°

adjacent nonadjacent

50°40°

Do complementary angles have to be adjacent?

nonadjacentadjacent

105°75°

70° 110°

Do supplementary angles have to be adjacent?

3. Are the two angles formed complementary or supplementary?

120°60°

supplementary angles

Two angles are vertical angles if they are NOT adjacent and their sides are formed

by 2 intersecting lines.

21

34

vertical angles

1 and 2 3 and 4

Two angles form a linear pair

if they are adjacent angles whose noncommon sides

form a line.

21

34 1 and 4

2 and 4

linear pairs

1 and 3

2 and 3

Which angles are… supplementary? congruent?

2

13

4

1 and 4

1 and 3

2 and 4

2 and 3

1 and 2

3 and 4

Linear Pair Postulate

If two angles form a linear pair, then they are

_____________.supplementary

Vertical Angles Theorem

Vertical angles are ___________.congruent

4. Find the measure of each angle.

21

340°

m1 = 140°

m2 = 140°

m3 = 40°

5. Find x and y. Then find the angle measures.

(4x + 15)° (3y + 15)°

(3y - 15)°(5x + 30)°

5x+30+4x+15=180

y = 30

105°

105°75°

75°

x = 15

3y-15+3y+15=180

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Lesson 1.7Perimeter

& AreaYou will learn to…* find the perimeter and area of

common plane figures

AreaHow many squares will

fit inside a region?

PerimeterWhat is the distance

around a region?

Square Area =

Perimeter =

side2

A = s2

4(side) P = 4s

RectangleArea =

2(base) + 2(height)

P = 2b + 2h

base ( height )

A = b h Perimeter =

1. Find the area of a square that has a side length of 20 inches.

Find the perimeter.

A = 202

A = 400 inches2

P = 4(20)P = 80 inches

2. Find the area of a rectangle that is 20 m by 4 m. Find the perimeter.

A = 20(4)A = 80 m2

P = 2(20) + 2(4)P = 48 m

3. A rectangle has an area of 98 cm2. Find the length of its base if its height is 7 cm.

A = b(h)98 = b(7)b = 14 cm

TriangleArea =

sum of 3 sides

P= a + b + c

½ base ( height)

A=½ b hPerimeter =

Find the area and perimeter of the triangle.4.

16 cm

9 cm15 cm

9(16) 2A = = 72 cm2

P = 40.8 cm

9.8 cm

c2 = 42 + 92

124152 = x2 + 92

5. Find the area and perimeter of the triangle.

9 cm

10 cm6 cm

A = 9(6) 2

A = 27 cm2

?

c2 = 92 + 102

13.5

c2 = 92 + 42

?9.8

P = 29.3 cm

Find the area of the triangle.

6.

20 cm

12 cm

25 cm 25(12) 2 A=

A = 150 cm2

Find the area of the triangle.

6.

20 cm

12 cm

25 cm 20(15) 2 A=

A = 150 cm2

Pythagorean Theorem?

15 cm

radiusdiameter

center

2 (radius)diameter =

Circumference of a Circle

C = (diameter) πC = (2 · radius) π

C= dπ or C=2rπ

7. Find the circumference.

8. Find the circumference.

6 cm

C = 37.7cm

C =10 cm

31.4 cm

12π cm ≈

10π cm ≈

2π 2π

9. Find the radius of the circle that has a circumference of 120 feet.

120 = 2rπ

19.1≈ r

C = dπ C = 2rπ C = 2rπ

Area of a Circle

A = (radius) 2 π

A = r 2 π

10. Find the area.

11. Find the area.

6 cm

A =

A =16 cm

A ≈

A ≈

36π cm2

113.1 cm2

64π cm2

201.1 cm2

12. Find the radius of the circle that has

an area of 120 square feet.120 = r2 π

r ≈ 6.2 ft

π π

r2 = 38.197

r = 38.197

13. Find the perimeter and area of the triangle.

AC = 6

AB =

BC =

34 ≈ 5.8

34 ≈ 5.8

P = 6 + 2 34 ≈ 17.6

A = ½ (5)(6) =(-2,-4)

A(3,-1)B

(-2,2)C

height = 5

15 square units

AB = (-2 - 3)2 + (-4 - -1)2BC = (3 --2)2 + (-1 - 2)2

14. Find the area of the bluish region.

Square - Circle = 144 - 36π ≈ 30.9 cm2

Square?

12 cm Circle?A = 62 π = 36π

A = 122 = 144

15. Find the area of the bluish region. r = 5

Square - Circles = 400 – 4(25π) ≈ 85.84

Square?

Circle?A = 52 π = 25π

A = 202 = 400


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Lesson 1.1 Patterns and Inductive Reasoning