Download Unit 3 Lesson 1 Basic Definitions

Objectives:
I can Identify and
model points, lines,
planes, and angles.
I can Identify
intersecting lines and
planes.
Basic
Geometry
Review
Unit 3
Lesson 1
We will be identifying basic
geometric terms.
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
A
B
n
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
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.
X 
Z
Y
T
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.
10
Different planes in a figure:
A
D
B
C
E
H
Plane EFGH
F
G
Plane ABCD
Plane BCGF
Plane ADHE
Plane ABFE
Plane CDHG
Etc.
Lesson 1-1 Point, Line, Plane
 Points
that lie on the same plane are said to be
coplanar.
A
B
D
C
M
 Noncoplanar
plane.
points are points not in the same
K
B
A
F
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 C
3. Name three points that
A
are collinear.
l
E
D
C
B
N
4. Are points E, A, B, and D collinear or non
collinear?
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
Intersection is the set of points in both figures.
Lines intersect at a point.
k
j
P
Example
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.
A
E
P
C
D
B
Example
Choose 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.
B.
C.
D.
A
line and a plane intersect at a point.
K
P
 Planes
intersect at a line.
K
B
A
F
True or False
1.
Line PF ends at P.
false
2.
Point S is on an infinite number of lines.
true
3.
The edge of a plane is a line.
false
Interpret Drawings
Example 1
A. How many planes appear in this figure?
Answer: There are two planes: plane S and plane
ABC.
Interpret Drawings
Example 1
B. Name three points that are collinear.
Answer: Points A, B, and D are collinear.
Interpret Drawings
Example 1
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.
Interpret Drawings
Example 1
Answer: The two lines intersect at point A.
1)
A. point X
B. point N
A.
C. point R
B.
C.
D. point A
D.
A
B
C
D
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.
31
Congruent Segments
Definition: Segments with equal lengths. (congruent symbol:

)
B
Congruent segments can be marked with dashes.
A
If numbers are equal the objects are congruent.
C
AB: the segment AB ( an object )
AB: the distance from A to B ( a number )
Correct notation:
AB = CD
AB  CD
Incorrect notation:
AB  CD
AB = CD
D
Congruent Segments
Segment Bisector
Definition: Any segment, line or plane that divides a segment into
two congruent parts is called segment bisector.
A
F
A
B
E
AB bisects DF.
E
AB bisects DF.
B
D
F
A
E
D
D
F
Plane M bisects DF.
B AB bisects DF.
Segment Bisector
Any segment line or plane that intersects a segment
at it’s midpoint.
n
4
A
X is the midpt of AB
4
X
B
AX  XB
n bisects AB
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?
Ray
Definition: RA : RA and all points Y such that
A is between R and Y.
A
How to sketch:
R
A
Y
R
How to name:
RA ( not AR )
RA or RY ( not RAY )
( the symbol RA is read as “ray RA” )
Opposite Rays
Definition: If A is between X and Y, then ray AX and
ray AY are opposite rays.
( Opposite rays must have the same “endpoint” )
X
A
Y
opposite rays
D
not opposite rays
E
DE and ED are not opposite rays.
Angles
 An Angle
is a figure formed by two rays with a common endpoint,
called the vertex.
ray
vertex

ray
Angles can have points in the interior, in the exterior or on the angle.
E
A
D
B
C
Points A, B and C are on the angle. D is in the interior and E is in the
exterior.
B is the vertex.
Naming an angle
(1) Using 3 points
(2) Using 1 point
(3) Using a number – next slide
Using 3 points: vertex must be the middle letter
This angle can be named as ABC or CBA
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 B .
B
A
C
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
A
number is placed in the interior of the angle near
its vertex. The angle to the left can be named
B
2
C
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.
Example

K is the vertex of more than one angle.
Therefore, there is NO K in this diagram.
There is LKM , PKM , and LKP
There is also 2 and 3 but there is no 5!!!
L
M
2
K
3
P
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
l
Just as we can measure segments, we can also measure angles.
l
We use units called degrees to measure angles.

A circle measures _____
360º
?

A (semi) half-circle measures _____
?

?
A quarter-circle measures _____
90º

One degree is the angle measure of 1/360th of a circle.
Adding Angles
l
l
When you want to add angles, use the notation m1,
meaning the measure of 1.
If you add m1 + m2, what is your result?
m1 + m2 = 58.
A
B
36°
m1 + m2 = mADC also.
22°
Therefore, mADC = 58.
1
D
2
C
Angle Bisector
An angle bisector is a ray in the interior of an angle that splits the
angle into two congruent angles.
Example: Since 4   6, UK is an angle bisector.
41° K
j
4
U
6
41°
5
3
Example
 Draw
your own diagram and answer this question:
 If ML is the angle bisector of PMY and mPML = 87,
then find:
 mPMY = _______
 mLMY = _______
Congruent Angles
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:
3   5.
3
5
Example
 Draw
your own diagram and answer this question:
 If ML is the angle bisector of PMY and mPML = 87,
then find:
 mPMY = _______
 mLMY = _______
Practice
Name all angles that have B as a vertex.
A
G
B
D
E
C
F