Points, Lines, Planes

1-3

Quiz

• What is a conjecture?• Give a counterexample: The product of 2

positive numbers is greater than either number.

• Use inductive reasoning: 2, 6, 7, 21, 22, 66 67, ….

• Make a conjecture about the sum of the first 500 positive even numbers.

Collinear and Noncollinear Points

• Points that are on the same line are called collinear points.

• If a single line cannot be drawn through all the points, then the points are noncollinear.

C D E F

Collinear points

UV

WX

Noncollinear points

Collinear Points

E

C

D

P

F

mn

l

Are points E, F, and C collinear?Are points E, F, and D collinear?Are points F, P, and C collinear?Can you label line m in three ways?

Planes

Intersecting lines are lines that cross (intersect) at exactly one point.

Parallel lines are lines that do not cross They have no points in common.

Skew lines are not parallel and they do not intersect. They lie in different planes.

Planes

E

H G

F

C

BA

List three different names for the plane represented by the top of the box.

Basic Postulates of Geometry

• Postulate 1-1: – Through any two points there is exactly

one line.

Basic Postulates of Geometry

• Postulate 1-2– If two lines intersect, then they intersect

in exactly one point.

Basic Postulates of Geometry

• Postulate 1-3– If two planes intersect,

then they intersect in

exactly one line.

T

S

R

W

Basic Postulates of Geometry

• Postulates1-4– Through any three noncollinear points

there is exactly one plane.

Homework Prentice Hall Geometry

•Page 19- 20

– # 2- 32 even

•16 problems

Angles

QUIZ

• What is the answer to number…..– 6– 8– 14– 20– 28– 30

Segments

A segment is a part of a line with two endpoints.

R B

Read it as: “Segment RB” or “Segment BR”

Write it as:

RB or BR

Rays

A ray consists of an endpoint and all the points of a line on one side of the endpoint.

C D

Read it as: “Ray CD” (the order does matter)

Write it as:

CD

Vocabulary

• An angle has two sides and a vertex.

• The sides of the angles are rays. The rays share a common endpoint (the vertex)

• Angles are measured in units called degrees.

Right Angles

• Forms a square corner• Forms a 90 degree angle.

90 degrees

Straight Angle

• Forms a straight line• Angle is 180 degrees

180 degrees

Types of Angles

When lines intersect to form right angles, then they are classified as perpendicular lines.

What do we use to help us?

A protractor

Here is a standard protractor like you use in the classroom.

When we use a protractor, we need to line it up correctly.

You need to make sure the protractor is lined up correctly.

Is this ready to measure the angle?

Were you right.................it wasn’t

Look for the upside down ‘T’ in the middle of the straight line on your protractor.

This needs to be exactly on the vertex of your angle.

We need to remember.....

It doesn’t matter which way round the angle is, you ALWAYS need to line the upsidedown ‘T’ to the vertexof the angle.

Now you are ready.

Read from the 0°, and follow the inner set of numbers.

Homework

•Page 25•# 1-15

Homework Prentice Hall Geometry

QUIZ

1. When two lines are skew, it means three things. What are those three things?

2. Demonstrate how to label a line, plane, ray, and segment.

3. An angle is made up of two ____ sharing a common point called _____.

4. The term skew is a Middle English word meaning "to escape."

Skew lines cannot be contained in one plane. Therefore, they have "escaped" a

plane.

What is something in your life that you have skewed from, • for the better?

Labeling Angles

C D

B

Must have three points. One on each ray and the vertex.

Labeled as: ∠ BCD or ∠DCB

Complementary Angles

If the sum of the measures of two angles is exactly 90º then the angles are complementary.

Supplementary Angles

If the sum of the measures of two angles is exactly 180º then the angles are supplementary.

Vertical Angles

Two angles whose sides are opposite rays.

Vertical Angles

Vertical angles are congruent

Congruent means same or equal.

In this picture, <1 and <3 are vertical angles.

What is the measure of <2?

<2 is 120º because it is congruent to the vertical angle across from it.

Adjacent Angles

Two coplanar angles with a common side, a common vertex, and no common interior points.

A

B

C

D

∠ABC is adjacent to ∠CBD

Review! Name each picture.

Postulates.Angle Addition Postulate

If point B is in the interior of ∠AOC, then m ∠AOB + m ∠BOC = m ∠AOC.A

O

B

C

Postulates.Angle Addition Postulate

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

AO

C

B

Homework Prentice Hall Geometry