Basic GeometryReviewUnit 3Lesson 1

We will be identifying basic geometric terms.

Objectives:

I can Identify and model points, lines, planes, and angles.

I can Identify intersecting lines and planes.

In geometry, the terms point, line, and plane are considered undefined terms. They are explained through examples and descriptions, not definitions.

A point is simply a location.

P

Drawn as a dot Named by a capital letter Has neither shape nor size called “point P”

A line is made up of an infinite set of points extending in two directions.

A line has no thickness or width, but it does have length.

A B

A line has an arrowhead at each end A line can be named by a lowercase letter OR if

two points are known, then the line can be named by those letters.

Called line n, line AB or AB, line BA or BA We can also represent a line like: AB

A B n

a LINE

Points on the same line are said to be collinear.

A B

Noncollinear points are points NOT on the same line.

A B

C

A plane is a flat surface made up of an infinite set of points that creates a

flat surface that extends without ending.

A plane has an infinite length and width, but no depth.

X Y

Z T

Drawn as a shaded, slanted 4-sided figure Named as a capital letter or by using three

non collinear points on that plane. What could we name this plane?

Plane T, plane XYZ, plane XZY, plane YXZ, plane YZX, plane ZXY, plane ZYX.

X Y

Z T

a PLANE

Lesson 1-1 Point, Line, Plane

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Different planes in a figure:A B

CD

EF

GH

Plane ABCD

Plane EFGH

Plane BCGF

Plane ADHE

Plane ABFE

Plane CDHG

Etc.

Points that lie on the same plane are said to be coplanar.

A BD

CM

Noncoplanar points are points not in the same plane.

F

K

B

A

Modeling Points, Lines, and Planes

Let’s take a look at a piece of paper

Naming Lines and Planes

1. Name a line containing point A.2. Name a plane containing

point C3. Name three points that

are collinear.

4. Are points E, A, B, and D collinear or non collinear?

E

l

D C

B A

N

FACTS It takes at least two points to make a line.

It takes at least three points to make a plane.

Space is the set of all points.

1. Are points A, B, and C collinear or noncollinear?

2. Are points B, C, and E collinear or noncollinear?

3. What are some ways to name this line?

A E

B C

Practice

Intersection is the set of points in both figures.Lines intersect at a point.

j k

P

Draw and label a figure for the following situation. Plane R contains lines AB and DE, which intersect at point P. Add point C on plane R so that it is not collinear with AB or DE.

Example

PA

D

E

B

C

A. B.

C. D.

ExampleChoose the best diagram for the given relationship. Plane D contains line a, line m, and line t, with all three lines intersecting at point Z. Also point F is on plane D and is not collinear with any of the three given lines.

A line and a plane intersect at a point.

P

K

Planes intersect at a line.

F

K

B

A

True or False

1. Line PF ends at P.

2. Point S is on an infinite number of lines.

3. The edge of a plane is a line.

false

true

false

Example 1

Interpret Drawings

A. How many planes appear in this figure?

Answer: There are two planes: plane S and plane ABC.

Example 1

Interpret Drawings

B. Name three points that are collinear.

Answer: Points A, B, and D are collinear.

Example 1

Interpret Drawings

C. Are points A, B, C, and D coplanar? Explain.

Answer: Points A, B, C, and D all lie in plane ABC, so they are coplanar.

Example 1

Interpret Drawings

Answer: The two lines intersect at point A.

A. AB. BC. CD. D

1)

A. point X

B. point N

C. point R

D. point A

2)

Draw a surface to represent plane R and label it.

ANSWER

2)

Draw a surface to represent plane R and label it.

Congruent refers to objects that have the same shape or size.

*Congruent segments are segments that have equal length!

When writing and signifying congruence, we use the ≅ symbol. When drawing a picture of figures

that are congruent, we use slashes or ticks.

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Congruent Segments

Definition:

If numbers are equal the objects are congruent.

AB: the segment AB ( an object )

AB: the distance from A to B ( a number )

AB

D

C

Congruent segments can be marked with dashes.

Correct notation:

Incorrect notation:

AB = CD AB CD

AB = CDAB CD

Segments with equal lengths. (congruent symbol: )

Congruent Segments

Segment BisectorAny segment, line or plane that divides a segment into two congruent parts is called segment bisector.

Definition:

B

E

D

FA

BE

D

FA

E

D

A F

B

AB bisects DF. AB bisects DF.

AB bisects DF.Plane M bisects DF.

Segment BisectorAny segment line or plane that intersects a

segment at it’s midpoint.

If X is between A and B and X is the midpoint of AB, what is the measure of AX if AB = 16x – 6 and XB = 4x + 9 ?

n

44

XA B

X is the midpt of AB

AX XB

n bisects AB

Ray

Definition:

( the symbol RA is read as “ray RA” )

How to sketch:

How to name:

R

A R A Y

RA ( not AR ) RA or RY ( not RAY )

RA : RA and all points Y such that A is between R and Y.

Opposite Rays

Definition:

( Opposite rays must have the same “endpoint” )

AX Y

D ED E

opposite rays not opposite rays

DE and ED are not opposite rays.

If A is between X and Y, then ray AX and ray AY are opposite rays.

Angles An Angle is a figure formed by two rays with a common endpoint,

called the vertex.

vertex

ray

ray Angles can have points in the interior, in the exterior or on the angle.

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

A

BC

DE

(1) Using 3 points (2) Using 1 point (3) Using a number – next slide

ABC or CBA

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

B

Naming an angle

Naming an Angle - continued

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 .2

* 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

32

K

L

M

P

Therefore, there is NO in this diagram.There is , ,LKM PKM and LKP

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

K is the vertex of more than one angle.

4 Types of Angles

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 between 90 and 180.

Straight Angle:an angle that is exactly 180 .

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 (semi) half-circle measures _____

A quarter-circle measures _____

One degree is the angle measure of 1/360th of a circle.

?

?

?

360º

180º

90º

Adding Angles

When you want to add angles, use the notation m1, meaning the measure of 1.

If you add m1 + m2, what is your result?

m1 + m2 = 58.

22°

36°

21

D

B

C

A

Therefore, mADC = 58.

m1 + m2 = mADC also.

Angle Bisector

An angle bisector is a ray in the interior of an angle that splits the angle into two congruent angles.

j41°

41°

64

U

K53

Example: Since 4 6, is an angle bisector.

Example

Draw your own diagram and answer this question:If is the angle bisector of PMY and mPML = 87,

then find:mPMY = _______mLMY = _______

ML

3 5.

Congruent Angles

53

Definition: If two angles have the same measure, then they are congruent.

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

The symbol for congruence is

≅

Example:

Example

Draw your own diagram and answer this question:If is the angle bisector of PMY and mPML = 87,

then find:mPMY = _______mLMY = _______

ML

PracticeName all angles that have B as a vertex.

CF

E

D

A

BG