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### Section One

### Section 1.6 endpoint. The endpoint is the vertex of the angle.

Patterns and Inductive Reasoning

Inductive Reasoning

- Reasoning that is based on patterns you observe.

Find a pattern:

3,6,12,24,…

Find a pattern:

384,192,96,48,…

Conjecture

- A conclusion you reach using inductive reasoning

- Make a conjecture about the sum of the first 30 odd numbers:
- = 1
- 1 + 3 = 4
- 1 + 3 + 5 = 9

12

22

32

You can conclude that the sum of the first 30 odd

numbers is 30² = 900

Conjecture

- A conclusion you reach using inductive reasoning

Make a conjecture about the sum of the first 8 even numbers:

2 = 1 * 2

2 +4 = 2 * 3

2 + 4 + 6 = 3 * 4

Therefore, the first 8 even numbers = 8 * 9 = 72

You can conclude that the sum of the first 50 even

numbers is 50 * 51 = 2550

The sum of two numbers is greater than either number

8 + (-5) = 3 3 > 8

Counterexample- Not all conjectures turn out to be true.
- If you can find one example that proves the conjecture false, then you have found a counterexample

Closure

- How can you use a conjecture to help solve a problem?

A conjecture can be tested to see whether it is a solution

How many lines can you draw?

Mark 3 points. How many lines can you

draw?

3 Lines

Mark 4 points. How many lines can you

draw?

6 Lines

Mark 5 points. How many lines can you

draw?

10 Lines

Mark 6 points. How many lines can you

draw?

15 Lines

Definitions

Point – has no size; it is a location represented by a small dot and is named by a capital letter ( a geometric figure is a set of points)

Line – a series of points that extends in two opposite directions without end

Collinear – points that lie on the same line

C •

A •

B •

Plane – is a flat surface that has no thickness. A

plane contains many lines and extends without end

in the directions of its lines

Plane D

Coplanar – points and lines in the same plane

Since it takes 3 points to define a plane, 3 points are always coplanar. Unlike 4 or more points are sometimes coplanar.

Plane ABC

Postulate (or axiom)- an accepted

statement of fact

Postulate 1-1: Through any two points is exactly one line

Line t is the only line that passes through points A and B

t

•

B

•

A

Postulate 1-2 : If two lines intersect, then they intersect in exactly one point

•

•

B

E

AE and BD intersect at C

C

•

•

A

D

Postulate 1-3: If two planes intersect,then they intersect in exactly one line

R

•

S

•

T

•

W

Plane RST and Plane STW intersect

in line ST

Postulate 1-4: Through any three noncollinear

points there is exactly one plane

Closure

How are a postulate and conjecture alike?

You accept a postulate as true without proof

How are a postulate and conjecture different?

You try to determine whether a conjecture is true or false

Segment AB

Segment – the part of a line consisting of two endpoints and all points between them

A

B

Ray – part of a line consisting of one endpoint and all the points of the line on one side of the endpoint

Ray XY

X

Y

Opposite Rays – two collinear rays with the same endpoint. Opposite rays always form a line.

ML and MN are opposite rays

L

M

N

Skew lines –are not parallel and do not intersect, therefore, they are noncoplanar

Parallel lines – are coplanar lines that do not intersect

E

AB and CD are parallel lines

AB and EF are skew lines

A

B

F

C

D

Parallel Planes – are planes that do not intersect therefore, they are noncoplanar

L

A

P

Y

C

I

N

E

Plane PLAY is parallel with plane NICE

Closure therefore, they are noncoplanar

How are parallel lines and skew lines alike?

How are parallel lines and skew lines different?

Both parallel lines and skew lines never intersect

Parallel lines are coplanar and skew lines are noncoplanar

Postulate 1-5 therefore, they are noncoplanar

The points of a line can be put into one-to-one

Correspondence with the real numbers so that the

distance between any two points is the absolute

value of the difference of the corresponding

numbers

a

b

A

B

The length of AB = a - b

Two segments with the same length are therefore, they are noncoplanar

CONGRUENT SEGMENTS

A

B

C

D

AB = CD therefore AB CD

Postulate 1-6 therefore, they are noncoplanar

If three points A, B, and C are collinear and B is

between A and C, then AB + BC = AC

A

B

C

Midpoint – of a segment is a point that divides a segment into two congruent segments; it bisects the segment

- Angle – an angle is formed by two rays with the same endpoint. The endpoint is the vertex of the angle.

A

B

C

Ray BA and Ray BC for angle ABC

B is the endpoint. B is the vertex of ABC

Postulate 1-7: endpoint. The endpoint is the vertex of the angle.Protractor Postulate

Let OA and OB be opposite rays in a plane. OA, OB, and all the rays with endpoint O that can be drawn on one side of AB can be paired with the real numbers from 0 to 180 so that

a. OA is paired with 0 and OB is paired with 180

b. If OC is paired with x and OD is paired with y, then

m COD = |x - y|

Classify angles to their measures endpoint. The endpoint is the vertex of the angle.

Acute angle

0 < X < 90

Right Angle

X = 90

Obtuse Angle

90 < X < 180

Straight Angle

X = 180

Postulate 1-8: Angle Addition endpoint. The endpoint is the vertex of the angle.

If point B is in the interior of AOC, then

m AOB + m BOC = m AOC

If AOC is a straight angle, then m AOB + m BOC = 180

A

B

O

C

B

O

A

C

Closure endpoint. The endpoint is the vertex of the angle.

- How can two of the postulates in this lesson help you measure segments and angles?

The segment add. Postulate states that the sum of all the parts of a segment equals the whole segment. Angle add. Postulate states that an angle can be divided into two angles, the sum of whose measures equals the measure of the whole angle

Definitions endpoint. The endpoint is the vertex of the angle.

- Perpendicular lines – two lines that intersect to form right angles.

- Perpendicular bisector of a segment – is a line, segment, or ray that is perpendicular to the segment at its midpoint, bisecting the segment into two congruent segments

- Angle bisector – a ray that divides an angle into two congruent coplanar angles. Its endpoint is at the angle vertex.

The Coordinate Plane

A endpoint. The endpoint is the vertex of the angle.

B

Distance FormulaThe distance dbetween two points A(x₁, y₁) and B(x₂, y₂) is calculated using the formula below:

d = (x₂ – x₁)² + (y₂ – y₁) ²

(x1 , y1)

(x2 , y2)

d = endpoint. The endpoint is the vertex of the angle. ((x₂ – x₁)² + (y₂ – y₁)²)

d = ((-8 – 10)² + (14 – 14 )²)

d = ((-18)² + (0)²)

d = 324 + 0

ExamplesL (10,14) M (-8, 14)

d = 18

d = endpoint. The endpoint is the vertex of the angle. ((x₂ – x₁)² + (y₂ – y₁)²)

d = ((-8 – -3)² + (6 - -2)²)

d = ((-5)² + (8)²)

d = 25 + 64

ExamplesL (-3, -2) M (-8, 6)

d = 9.43

The Midpoint Formula (m endpoint. The endpoint is the vertex of the angle.x, my)

The (x, y) coordinates of the midpoint M, of line segment AB whose endpoints are A(x₁, y₁) and B(x₂, y₂), are found using this formula

Coordinates of Mx and My are:

x₁+ x₂ , y₁ + y₂

2 2

Coordinates of M endpoint. The endpoint is the vertex of the angle.x and My are:

Mx = x₁+ x₂ and My = y₁ + y₂

2 2

If given one endpoint and the midpoint, we can find the (x,y) coordinates of the other endpoint by simplifing this formula into:

Mx = x2 + x1 My = y2 + y1 2 2

2Mx = x2 + x1 2My = y2 + y1

2Mx – x1 = x2 2My – y1 = y2

Closure endpoint. The endpoint is the vertex of the angle.

If you know the coordinates of the midpoint and one endpoint, can you find the coordinates of the other endpoint?

2Mx – x1 = x2 2My– y1 = y2

Midpoint: (2,5)

Endpoint A (-1,5)

Endpoint B (x, y)

d = endpoint. The endpoint is the vertex of the angle. ((5 - -6)² + (-1 - -6)²)

d = ((11)² + (5)²)

d = (121 + 25)

d = 146

Finding the distance and midpoint of a geometric figure(-2 , 8)

d2

d3

d1 = 12.08

(5, -1)

d1

d2 = 11.40

(-6, -6)

d3 = 14.56

Perimeter endpoint. The endpoint is the vertex of the angle.

- The perimeter of a polygon is the sum of the lengths of its sides

Perimeter of a square = side + side + side + side = 4s

S

If the side of a square is 8cm, what is the perimeter?

S

S

S

Perimeter endpoint. The endpoint is the vertex of the angle.

- Perimeter of a rectangle =
base + base + height + height = 2b + 2h

B

If the base is 8 inches and the height is 4 inches, what is the perimeter?

H

H

B

Area endpoint. The endpoint is the vertex of the angle.

- The area of a polygon is the number of square units the figure encloses

Area of a square = (side)(side) = s

What is the area of the square???

4 ft

4 ft

Area endpoint. The endpoint is the vertex of the angle.

Area of a rectangle = (base)(height) = bh

12 in

3 ft

What is the area of the rectangle???

6 in endpoint. The endpoint is the vertex of the angle.

Circumference- This is referred to when measuring the perimeter of a circle

Perimeter of a circle = circumference of a circle

= πd (or 2πr )

What is the circumference of the circle???

Area of a circle endpoint. The endpoint is the vertex of the angle.

- Area = r2

What if we were given the diameter. How do we use this formula???

2 radii = 1 diameter

Area = r2= (d/2)2 = d2 4

Closure endpoint. The endpoint is the vertex of the angle.

- Can you find the area and perimeter of the square? Circle?

The sides of the square measure 8 cm

Square: Area = 64 cm² ; perimeter = 32 cm

Circle: Area = 32π cm² ; circumference = 82π cm

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