1.2 – Points, Lines, and Planes• Warm – Up, Do if you finished checking

homework

Find the slope of the line going through these two points.

Because I know some of you forgot:

• (4,5) (-1,3) (-3,1) (2,-5)

• Does order matter? When? Why?

Point. Technically, it has no size, but we use a dot that has size to represent it. You use a capital letter to label it. Such as Point AAll figures are made of points. This is a LINE. It goes both ways, forever without ending. Once again, it has no thickness, but we use a picture with thickness to describe it. Arrows on both ends say it goes on forever.

A definition uses known words to describe a new word. In geometry, are undefined terms.

PLANE, goes on forever, once again has no thickness. Even though it goes on forever, we usually use a parallelogram shape to draw it.

To label it, a capital cursive letter can be used, or you can use three points that don’t line up (also known as non-collinear points)

A

K

I

M

Collinear points, points all in one line.

Noncollinear points, points NOT all in one line.

Noncoplanar points, points NOT all in one plane.

Coplanar points, points all in one plane.

D

UC

K

S

F BI

TR

T, R are ENDPOINTS

This is a line segment, it is a segment, or part of a line

TR

labeledisIt

This is a ray

BIorFI

labeledisIt

pointinitial

thecalledisF

FIrayFor

pointinitial

thecalledisB

BIrayFor

OPPOSITE RAYS –

are called opposite rays cuz N is between M and O.

M

l

NO

NMandNO

AD C

B

H

E

G

F

I

J

Q

P

l

Name four coplanar points

Which plane has points F,H,I?

Name three collinear points

What is the intersection of line l and plane P?

• Warm – Up, Do if you finished checking homework

• Solve for y. Remember, that means make it ‘y=‘

2x + 3y = 6 y – 3 = -2(x – 1)

What does collinear mean?Why do two points HAVE to be collinear?

ACBC,,AB Draw

C. and B, A, pointsar noncolline threeDraw

Two or more geometric figures intersect if they have one or more points in common. The intersection of the figures is the set of points the figures have in common.

Draw three noncollinear points A, B, C on plane P. Draw line l not on plane P going through point C.

Draw three planes M, N, P meeting at point P.

Draw three planes M, N, P meeting on line l.

In 3-D, sometimes it helps to imagine a box, or look around the room (but not during a test)

• Three lines l, m, n meet at point P, where lines m, n are on plane P.

Four points, A, B, C, D, where A, B, and C are coplanar but not collinear on plane P, and D is noncoplanar

Warm – Up: Find the distance between the points

A B

Where your ruler is doesn’t matter. Two points are the same distance apart no matter how you line up the ruler.

Two find the distance between two points, you can just subtract the distance and take the absolute value.

Like if on a number line, I have a points at Q and R, and they are different, to find the distance, all I have to do ___________ and then ____________

C D

E F

1.3 Segments and Their Measures

Postulate \ Axiom – A rule that is accepted WITHOUT PROOF.

Postulate 1 – 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 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.

0 1-1

BA

SEGMENT ADDITION POSTULATE

If B is BETWEEN A and C, then AB + BC = AC. Also

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

A B C

AC

AB BC

Why does this only work if it’s ___________?

B O X

______4_____ x

82 xxE A T

______50_____

82 xx4

D U C K S

DS = 30

DU = 5

KS = 7

UC = .5CK

UK =

UC =

DC =

US =

boardwork

T5

R H5

S

CONGRUENCEisThis

HSTR

EQUALisThis

HSTR

Congruence is shown with marks. The marks say that they are the same size and shape

Equals means they have equal length, number value.

They are equivalent. Definition of congruent segments: Congruent segments have equal lengths

Superposition:

(x1,y1)

(x2,y2)

What is Pythagorean Theorem?

What does a, b, and c represent?

Looking at the graph, one way to represent the length of the horizontal leg is:

Using the same logic, another way to represent the vertical leg is:

Replace a and b with what we just found and solve for c

FORMULADISTANCE(-5, -2) (4, 1)

x1 y1 x2 y2

Why do they use d instead of c?

How come order doesn’t really matter for this formula?

Why do you think they set it up this way?

(0, 0) (8, 0) (16, 0)

(0, 12)

(8, 6)

Find the distance between Mr. Kim and each food location.

From where Mr. Kim starts, if he goes to In-N-Out, Der Veener, and Carl’s, and back to where he started, how far does he walk?

What’s nice about finding distance when lines are horizontal or vertical?

• Warm – Up • Graph these lines. We will use them later.

• x = 0• y = 0• y = x

• x = 0 is also known as:

• y = 0 is also known as:

1.4 – Angles and Their Measures

A

N

G

LE

S

Angles are formed by two rays with the same initial point.Two rays are called the sides.

The initial point is called the vertex

1

21

21

mm

anglescongruentofDefinition

D

U

C

R

E

X

1 2

If two angles are congruent, their measures are equal.

If the measure of two angles are equal, they are congruent

How to use a protractor to measure angles.

Protractor Postulate

O B

A

Consider a point A on one side of OB. The rays of the form OA can be matched one to one with the real numbers from 0 to 180.

The measure of AOB is equal to the absolute value of the difference between the real numbers for OA and OB.

90

Acute – Angle is between __ and __ degrees

Right – Angle is exactly __ degrees

Obtuse – Angle is between __ and ___ degrees

Straight – Angle is ___ degrees

20 180 90 120

0180

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

A points is in the _______ of an angle if it is not on the angle or its interior

D

U

C

Let’s look at the warm-up and identify angels, interior, and exterior points.

Adjacent angles, share common side and vertex, but share NO interior points.

.

.

adjacentnotare

BUDandBUC

adjacentare

CUDandBUC

C

O

B

A

AOCmBOCmAOBm

thenAOC,

ofinteriortheinisBIf

PostulateAdditionAngle

In the future with proofs, angle and segment addition postulates will be important in putting together and breaking apart angles.

Find the measure of the unknown angles, state if they are acute, right, or obtuse.

A 23

4

D

CB

E

F

1

CAFm

DAFm

m

70

202

1

65

90

m

CAEm

BAEm

1 76o

Draw angle ABC that is 90o. Draw right angle DBF so that angle ABF and DBA is 45o and A is in the interior of angle DBF and F is in the interior of angle ABC.

FBCm

DBCm

DBAm

Find

• Draw a right angle KIM. Draw angle JIQ such that M is in the interior of angle JIQ and Q is in the interior of KIM and JIM is 30 degrees and MIQ is 60 degrees

(-3, -2) (5, -1)

x1 y1 x2 y2

Warm – Up: What coordinate is in the MIDDLE of these two points?

How did you find it?

What’s another way to think of the ‘middle’ of two numbers?

1.5 – Segment and Angle Bisectors

Find the midpoint.

(-2, -1) (2, 5) (5, -2) (3, 6)

A B C

SEGMENT BISECTOR – A line, segment, or ray that INTERSECTS THE _____________________________________!

The ___________ of a segment divides the segment into __________________parts.

D

E

ACoftmidpoinisB

BCAB

BCABtmidpoinofDefinition

:

ctorsbise

segmentare

etcBEDEDE ,,,

Given an endpoint and the midpoint, find the other endpoint. A is an endpoint, M is a midpoint

A (5, -2) M (3, 6) B (x, y) A (2, 6) M (-1, 4) B (x, y)

ANGLE BISECTOR – is a ray that divides an angle into two adjacent angles that are congruent.

BTRbisectsTA

ATRBTA

ATRmBTAm

bisectorangleofDefinition

B

A

R

T

2020

xfind ABC, bisects BD

A

D

C

B

A

D

C

B

5x

102

1x

42 x

67 x

Constructing a perpendicular bisector.

1) Point on one end, arc up and down.

2) Switch ends and do the same

3) Draw line through intersection

This is DIFFERENT from book (slightly).

Bisect an angle

1) Draw an arc going across both sides of the angle.

2) Put point on one intersection, pencil on other, draw an arc so that it goes past at least the middle.

3) Flip it around and to the same.

4) Line from vertex to intersection.

Square

A = P =

s

w

lRectangle

A = P =

A

trianglea of Area

Warm – Up: Things you should know from your past, fill in the blanks

Perimeter of a triangle, add up the sides

1.7 – Introduction to Perimeter, Circumference, and Area

Circumference is the distance around the

circle. (Like perimeter)

C = πd = 2πr

Area of a circle:

A = πr2

Find Perimeter\Circumference, and Area for each shape

3 in

6 ft

12 cm13 cm 15 cm

14 cm

5 ft

3 ft

8 cm12 cm 17 cm

Find the area and perimeter

Find the area of the figure described

Find the area of a circle with diameter 10 m

Find the area of a triangle with base 2 in and height 6 in

Find the area of a rectangle with base 4 ft and height 2 ft

Find the area of a square with perimeter 8 miles

Write on board

Finding Area

Mr. Kim needs to make a moat around his castle. The radius of the outer circle is 50 feet, the radius of the inner circle is 40 feet. What is the area of his moat?

How many square yards of flooring are needed to cover a room that is 18 ft by 21 ft?