lesson 1-1 point, line, plane modified by lisa palen

Lesson 1-1

Point, Line, Plane

Modified by Lisa Palen

1-1 Part A - Geometry :The Objects

Point, Line, Plane,Segment, Ray,

Angle

Undefined Terms

•Point•Line

•PlaneWe describe these, rather than defining them.

Point

• A place in space. Has no actual size.

• How to Sketch:

Use dots

• How to label:

Use capital printed letters

Never name two points with the same letter (in the same sketch).

A

B AC

B

Line

• Straight figure, extends forever, has no thickness or width.

• How to Sketch: Use arrows at both ends

• How to label:

(1) Use small script letters – line n(2) Use any two points on the line -

• Never name a line using three points.

Never name two points with the same letter (in the same sketch).

n

A

C

Plane

• Flat surface that extends forever in all directions.

• How to sketch: Use a parallelogram (four sided figure)

• How to name: 2 ways:

(1) Use a capital script letter – Plane M(2) Use any 3 noncollinear points in the plane

AB

C

Horizontal Plane

M

Vertical Plane Other

ABC

ACB

BAC

BCA

CAB

CBA

More Objects

•Segment •Ray

•Angle

Segment

part of a line that includes two points (called the endpoints) and all points between them

How to sketch:

How to name:

Definition:

AB or BA

The symbol AB is read as "segment AB".

AB (without a symbol) means the length of the segment or the distance between points A and B.

AB

Ray

Definition:

( the symbol RA is read as “ray RA” )

How to sketch:

How to name: RA or RY ( not RAY )

Part of a line starting at one point (called the endpoint)And extending forever in one direction.C

D

RA

Y

What is

?

Angle

vertex

ray

ray

Definition: Angle - Figure formed by two rays with a common endpoint, called the vertex. The two rays are called sides of the angle

Naming an angle: (1) Using 3 points (2) Using 1 point

(3) Using a number – next slide

ABC or CBA

B

Using 3 points: vertex must be the middle letter

This angle can be named as

Using 1 point: using only vertex letter

* Use this method is permitted when the vertex point is the vertex of one and only one angle.

Since B is the vertex of only this angle, this can also be called .

A

B C

Naming an Angle - continued

2

Using a number: A number (without a degree symbol) may be used as the label or name of the angle. This number is placed in the interior of the angle near its vertex. The angle to the left can be named

as .

* The “1 letter” name is unacceptable when …more than one angle has the same vertex point. In this case, use the three letter name or a number if it is present.

2

A

B C

Example

K, ,LKM PKM and LKP

32

K

L

M

P

Therefore, there is NO in this diagram.There are .

2 3 5!!!There is also and but there is no

K is the vertex of more than one angle.

Lesson 1-1 Part B VocabularyCollinear, Coplanar,

Intersection, Intersect, Parallel, Perpendicular


Collinear Points

Collinear points are points that lie on the same line. (The line does nothave to be visible.)

A

BC

Points A, B and Care noncollinear

Points A, B and C are collinear.

A B C

Definition:

Coplanar

Coplanar objects (points or lines) are objects that lie on the same plane. (The plane does not have to be visible.)

Definition:

QR

S

PP, Q, R and S are coplanar.

AD

C

B

A, B, C and D are noncoplanar.

Coplanar

Coplanar objects (points or lines) are objects that lie on the same plane. (The plane does not have to be visible.)

Definition:

H

E

G

DC

BA

F

Are they coplanar?

ABC ? yesABCF ? NOHGFE ? yesDCEF ? yesAGF ? yesCBFH ? NO

Line g and line r intersect at point I.

Intersection / Intersect

• Definition The intersection of two objects is the set of points in common to both objects. (where the objects touch.)

• Definition Two objects intersect if they have points in common. (if the objects touch.)

r

I gThe intersection of line g and line r is point I.

Intersection of Two Lines

• If two lines intersect, what is their intersection?

• Otherwise, they are either parallel or skew.

Intersectionis a point.

parallel

skew

• Two coplanar lines that don’t intersect• Symbol: means “║ is parallel to”

Parallel Lines

Parallel lines go in

the same direction.

w

v

v║w

Perpendicular Lines

• lines that intersect at right angles

• Illustration:

• Symbol: means “is perpendicular to”

• Key Fact: 4 right angles are formed.

m

n

mn

Lesson 1-2Segments and Rays


Recall: What is a Segment?

two points (called the endpoints) and all points between them

How to sketch:

How to name:

Definition:

AB or BA

The symbol AB is read as "segment AB".

AB

Measure (of a Segment)

The length of the segment or the distance between the two endpoints

Notation:

Definition:

AB

Recall: The symbol is read as “segment A B”.

AB (without a symbol) means the length of the segment or the distance between points A and B.

The measure of is AB.

Congruent Segments

Definition: Congruent segments are segmentswith equal measures (lengths).

AB

D

CMark congruent segments with . . dashes..

Congruent segments have the same number of dashes.

E F

HG

Notation: The symbol means “is

congruent to”.

CDAB GHEF EFAB

Congruent Segments

Using the Notation: AB

D

C

Numbers are equal. Objects are congruent.AB: the distance from A to B ( a number )AB: the segment AB ( an object )

Correct notation:

Incorrect notation:

AB = CD AB CD

AB = CDAB CD

Midpoint

A midpoint is a point that divides a segment into two congruent segments.

Definition:

EDF

and DE = EF

E is the midpoint of .

Segment Bisector

A segment bisector is ANY object that divides a segment into two congruent segments.

Definition:

B

E

D

FA B

E

D

FA

E

D

A F

B

AB bisects DF. AB bisects DF.

AB bisects DF.

Plane M bisects DF.

Postulates

Definition: a statement we accept as true without proof.

Examples:• Through any two points there is exactly one line.

• Through any three non-collinear points, there is exactly one plane.

Postulates

• If two lines intersect, then the intersection is a point.

Examples:

• If two planes intersect, then the intersection is a line.

The Ruler Postulate

The points on any line can be paired with the real numbers in such a way that:

• Any two chosen points can be paired with 0 and 1. • The distance between any two points in a number line is the absolute value of the difference of the real numbers corresponding to the points.

.

The Ruler Postulate says you can use a ruler to measure the distance

between any two points! (It also gives us a

formula.)

The Ruler Postulate

• So, we can measure the distance between two pointsusing a “ruler”.

PK = (distance is always positive)

| 3 - -2 | = 5

•Formula: take the absolute value of the difference of the two coordinates a and b: │a – b │

Reminder

• The coordinates are the numbers on the ruler or number line!

• The capital letters are the names of the points.

• Coordinates: -3, -2, -1, 0, 1, 2, 3, etc.• Points: G, H, I, J, etc.

-5 5

SRQPOLKJIHG M N

Another Example

Find the distance between I and S.

-5 5

SRQPOLKJIHG M N

Coordinate of I:Coordinate of S: 6

-4

│ -4 - 6 │= │ - 10 │ = 10

Take the absolute value of the difference: │a – b │

Finding the Midpoint(of Two Points on a Number Line)

a b

2

The coordinate of a midpoint of a segment whoseendpoints have coordinates a and b is

-5 5

SRQPOLKJIHG M N

Example

a b 3 ( 2) 10.5

2 2 2

Find the coordinate of the midpoint of the segment PK.

-5 5

SRQPOLKJIHG M N

Now find the midpoint on the number line.

So what do we mean by between?

Which picture shows, “C is between A and B”?

So “C is between A and B” means that C is ON the

segment .AB Okay, but this is not the

definition.

Between

Definition: •If C is between A and B, then AC + CB = AB.•If AC + CB = AB, then C is between A and B.

AC + CB = AB AC + CB > AB

This is also called the Segment Addition Postulate.

or The Segment Addition Postulate

between not between

The Segment Addition Postulate(This is the same as “between.” )

In OtherWords:

The whole is the sum of the parts.

Or: Part + Part = Whole

These are the same length.

The Segment Addition Postulate

AB

C

Example: If C is between A and B, AC = 4 and CB = 8, then find AB.

AC + CB = AB

4 + 8 = AB

84

AB

Step 1: Draw.Step 2: Label.Step 3: Find equation. (Substitute)Step 4: Solve.Step 5: Make sure you answer the question.

12 = ABPart + Part = Whole

The Segment Addition Postulate

Example: If E is between D and F, DE = 5 and DF = 15, then find EF.

DE + EF = DF

5 + EF = 15

EF5

15

Step 1: Draw.Step 2: Label.Step 3: Find equation. (Substitute)Step 4: Solve.Step 5: Make sure you answer the question.

EF = 10Part + Part = Whole

Midpoint

If E is the midpoint of , and DE = 5, then find EF and DF.

Example:

EDF

Step 1: Draw.Step 2: Label.Step 3: Find equation. (Substitute)Step 4: Solve.Step 5: Answer question.

5

DE = EF

10 = DF Part = Part

5 = EF 5 + 5 = DF

DE + EF = DF

Part + Part = Whole

5

Lesson 1-4

Angles

Angle

vertex

ray

ray

Definition: Angle - Figure formed by two rays with a common endpoint, called the vertex. The two rays are called sides of the angle

Angles and Points• Angles can have points

in the interior, in the exterior, or on the angle.

B

A

C

D

E

Points A, B and C are on the angle, D is in the interior and E is in the exterior. B is the vertex.

B

A

C

D

E

Interior / Exterior of an AngleDefinition (you don’t need to memorize this.)A point is in the interior of an angle if it does not lie on the angle itself and it lies on a segment whose endpoints are on the sides of the angle.

A, B, and C are on the angle.

An exterior point is a point that is neither on the angle nor in the interior of the angle.

Interior Point

Exterior Point

The Protractor Postulate

Given a ray AB and a number r between 0 and 180,there is exactly one ray with endpoint A extending to either side of AB, such that the measure of the angle formed is r degrees.

You don’t need to memorize this!

The Protractor Postulate says you can use a

protractor to measure angles!

The Ruler and Protractor Postulates

The Ruler Postulate lets us use a ruler to measure the distance between two points.

The Protractor Postulate lets us use a protractor to measure an angle..

Protractor Applet

http://www.keymath.com/DG/dynamic/protractor.html

Measuring Angles Just as we can measure segments, we can also

measure angles. We use units called degrees to measure angles.

– A circle measures _____

– A half-circle measures _____

– A quarter-circle measures _____

– One degree is the angle measure of 1/360th

of a circle.

?

?

?

360º

180º

90º

Measure (of an Angle)The size of the angle

Notation:

Definition:

The measure of ABC is

mABC

Angles are measured using units called degrees (in this class.)

A

CB

4 Types of Angles

Lesson 1-4: Angles 56

Acute Angle: an angle whose measure is less than 90.

Right Angle: an angle whose measure is exactly 90.

Obtuse Angle: an angle whose measure is greater than 90 and less than 180.

Straight Angle: an angle that measures exactly 180 .

A

B

C

D

Congruent Angles


2 4. 2

4

Definition: Congruent angles - angles that have equal measures

Congruent angles are marked with the same number of “arcs”.

The symbol for congruence is

Example:

is an angle bisector.

Since 3 5, bisects ABC .

Angle Bisector / Bisect

An angle bisector is a ray that splits the angle into two congruent angles. The ray bisects the angle.


BD��

j41°

41°

64

U

K 53

Example 1:

A

B

C

D

Example 2: UK��

Example 1 Angle BisectorIf is an angle bisector of PMY and

m PML = 68, then find:

•m PMY = _______•m LMY = _______

ML

Example 2 Angle BisectorIf is an angle bisector of PMY and

m PMY = 86, then find:

•m PML = _______•m LMY = _______

ML

Adding Angles When you want to add angle measures, use the

notation m1, meaning the measure of 1. If you add m1 + m2, what is your result?

22°

36°

21

D

B

C

AmADC = 36 + 22

mADC = 58

How did you know to add???

Angle Addition PostulateThat last example is an example ofThe Angle Addition Postulate:

If D is in the interior of ABC,

then m< ____ + m< ____ = m< _____ABD DBC ABC

If mABD + mDBC = m ABC,

then D is in the interior of ABC.

Angle Addition PostulateA simpler way to remember this postulate:

_______ + _______ = _________

part

part

whole part part whole

Lesson 1-5: Pairs of Angles 65

Lesson 1-5

Pairs of Angles


Adjacent Angles

A pair of coplanar angles with a common (shared) vertex and common side that do not have overlapping interiors. 1 and 2 are adjacent. 3 and 4 are not. 1 and ADC are not adjacent.

Adjacent Angles( a common side ) Non-Adjacent Angles

22°

36°

21

D

B

C

A4

3

Definition:

Examples:

http://www.mathwords.com/v/vertex.htm

http://www.mathwords.com/a/arm_angle.htm


Complementary Angles

A pair of angles whose sum of measures is 90˚Definition:

Examples:

1 and 2 are adjacent complementary angles.( have a common side )

21

Q

AB

C 1

2

Q

R

AB

F

G

m1 = 40°m2 = 50°

1 and 2 are complementary but not adjacent angles.( don’t have a common side )


Supplementary Angles

A pair of angles whose sum of measures is 180˚Definition:

Examples:

2 1

A Q

B

C

1

2

A QR

BF

G

m2 = 140°m1 = 40° 1 and 2 are adjacent

supplementary angles.

1 and 2 are supplementary but not adjacentangles.

Opposite Rays

AX Y

D ED E

opposite rays not opposite rays

DE and ED are not opposite rays.

Opposite Rays

Definition:Two rays with the same endpoint, that together form a line.

Or (better): Two rays with the same endpoint that together form a straight angle.

AX Y

AX and AY are opposite rays.

XAY is a straight angle


Linear Pair

A linear pair is a pair of adjacent angles whose non-adjacent rays form opposite rays.

2 1

A Q

B

C

m2 = 140°

m1 = 40°

Definition:

1 and 2 are a linear pair.

A linear pair is a pair of adjacent supplementary angles.

Another“Definition”:


Vertical Angles

A pair of non-adjacent angles formed by intersecting lines.

Definition:

Examples:

2 and 4

1 and 3

Another Definition: A pair of angles whose sides form opposite rays. The pairs of opposite rays are and &QA QC

��&QB QD

��

Postulates vs. Theorems

Definition: A postulate is a statement we accept as true without proof.

Examples: Segment Addition Postulate and Angle Addition Postulate

Definition: A theorem is a statement we use logic to show is true.

Examples: Linear Pair Theorem and Vertical Angles Theorem (next slides)

Lesson 1-5: Pairs of Angles

Theorem (Linear Pairs)A linear pair is supplementary.

2 1

A Q

B

C1 and 2 are supplementary.

Given:

Prove:

1 and 2 are a linear pair.

Statements Reasons

1. 1 & 2 are linear pair.2. and are opposite rays.3. AQC is a straight angle.4. mAQC = 1805. m1 + m2 = mAQC6. m1 + m2 = 180 7. 1 and 2 are supplementary.

QA QC1. Given2. Defn. linear pair3. Defn. opposite rays4. Defn. straight angle5. Angle Addition Postulate6. Substitution Property7. Defn. supplementary


Vertical Angles Theorem

Vertical angles are congruent.Theorem

Theorem: Vertical angles are congruent.

1. 1 & 2 and 2 & 3 are linear pairs

2. 1 & 2 and 2 & 3 are suppl.

3. m1 + m2 = 180, m2 + m3 = 1804. m1 + m2 = m2 + m35. m1 = m36. 1 3

1. Defn. linear pair/diagram2. Linear pairs are

supplementary.

3. Defn. supplementary4. Substitution Property5. Subtraction Property6. Defn. Congruent Angles

The diagramGiven:4

3

2

1A

Q

D

B

C

Prove: 1 3

Statements Reasons


What’s “Important” in Geometry?

360˚ 180˚ 90˚

4 things to always look for !

. . . and CongruenceMost of the rules (theorems)and vocabulary of Geometryare based on these 4 things.


Algebra and Geometry

( ) = ( )( ) + ( ) = ( )( ) + ( ) = 90˚( ) + ( ) = 180˚

Common Algebraic Equations used in Geometry:

If the problem you’re working on has a variable (x),then consider using one of these equations.


Example: If m4 = 67º, find the measures of all other angles.

3 4 180m m

3 67 180m

3 180 67 113m

4

3

2

1

67º

Step 1: Mark the figure with given info.

Step 2: Write an equation.

3 1 , . 3 1 117 Because and are vertical angles they are equal m m

4 2 , . 4 2 67 Because and arevertical angles they are equal m m


Example: If m1 = 23 º and m2 = 32 º, find the measures of all other angles.

4 23 ( 1 & 4 .)

5 32 ( 2 & 5 .)

m are vertical angles

m are vertical angles

6 5

4 3 2

1

Answers:

1 2 3 180

23 32 3 180

3 180 55 125

3 6 125

3 & 6 .

m m m

m

m

m m

are vertical angles


Example: If m1 = 44º, m7 = 65º find the measures of all other angles.

3 90m

1 4 44m m

4 5 90

44 5 90

5 46

m m

m

m

7

6 5 4

3

2 1

Answers:

6 7 90

6 65 90

6 25

m m

m

m

lesson 1-1 point, line, plane modified by lisa palen

Documents

noncollinear points

set of points

vertex point

vertex letter

angle figure

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angle continuedusing

use dotshow