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Unit 1
Learning Outcomes 1: Describe and Identify
the three undefined terms
Learning Outcomes 2: Understand Angle
Relationships
Part 1
Definitions:
Points, Lines and Planes
Undefined Terms
Points, Line and Plane are all
considered to be undefined terms.
– This is because they can only be explained
using examples and descriptions.
– They can however be used to define other
geometric terms and properties
A
<
B
>
Point
– A location, has no shape or size
– Label:
Line
– A line is made up of infinite points and has no thickness or width, it will
continue infinitely.There is exactly one line through two points.
– Label:
Line Segment
– Part of a line
– Label:
Ray
– A one sided line that starts at a specific point and will continue on
forever in one direction.
– Label:
F
<
E
A
B
>
Collinear
– Points that lie on the same line are said to be
collinear
– Example:
Non-collinear
– Points that are not on the same line are said to be
non-collinear (must be three points … why?)
– Example:
Plane
– A flat surface made up of points, it has no depth
and extends infinitely in all directions. There is
exactly one plane through any three non-collinear
points
Coplanar
– Points that lie on the same plane are said to be
coplanar
Non-Coplanar
– Points that do not lie on the same plane are said
to be non-coplanar
Intersect
The intersection of two things is the
place they overlap when they cross.
– When two lines intersect they create a
point.
– When two planes intersect they create a
line.
Space
Space is boundless, three-dimensional
set of all points. Space can contain lines
and planes.
Practice
Use the figure to give examples of the following:
1.
2.
3.
4.
Name two points.
Name two lines.
Name two segments.
Name two rays.
5.
6.
7.
8.
9.
Name a line that does not contain point T.
Name a ray with point R as the endpoint.
Name a segment with points T and Q as its endpoints.
Name three collinear points.
Name three non-collinear points.
QuickTime™ and a
decompressor
are needed to see this picture.
Part 2
Distance, Midpoint and Segments
Distance Between Two Points
Distance on a number line
• PQ = B  A or A  B
Distance on coordinate plane
– The distance d between two points with
coordinates x1, y1 and x2 , y2  is given by
d
x
2
 x1
  y
2
2
 y1 
2
Examples
Example 1:
– Find the distance between (1,5) and (-2,1)
Examples 2:
– Find the distance between Point F and
Point B
<
E
B
-6
-1
>
Congruent
When two segments have the same
measure they are said to be congruent
Symbol:
Example: A
B

>
<
<
>
D
C
AB CD
Between
Point B is between point A and C if and
only if A, B and C are collinear and
AB  BC  AC
<
A
B
C
>
Midpoint
Midpoint
– Halfway between the endpoints of the
segment. If X is the MP of AB then
AX  XB
<
A
X
B
>
Finding The Midpoint
Number Line
– The coordinates of the midpoint of a segment
whose endpoints have coordinates a and b is
ab
2
Coordinate Plane
– The coordinates of midpoint of a segment whose
endpoints have coordinates x1, y1 and x2 , y2 
are  x1  x2 , y1  y2 

2
2

Examples
The coordinates on a number line of J
and K are -12 and 16, respectively. Find
the coordinate of the midpoint of
Find the coordinate of the midpoint of
for G(8,-6) and H(-14,12).
Segment Bisector
A segment bisector is a segment, line or
plane that intersects a segment at its
midpoint.
Segment Addition Postulate
– if B is between A and C, then
AB + BC = AC
– If AB + BC = AC, then B is between
A and C
Part 3
Angles
Angle
An angle is formed by two non-collinear
rays that have a common endpoint. The
rays are called sides of the angle, the
common endpoint is the vertex.
Kinds of angles
Right Angle
Acute Angle
Obtuse Angle
Straight Angle / Opposite Rays
Congruent Angles
Just like segments that have the same
measure are congruent, so are angles
that have the same measure.
Angle Bisector
A ray that divides an angle into two
congruent angles is called an angle
bisector.
Angle Addition Postulate
– If R is in the interior of <PQS, then
m<PQR + m<RQS = m<PQS
– If m<PQR + m<RQS = m<PQS, then R is
in the interior of <PQS
Measuring Angles
How to use a protractor.
– 1.) Line up the base line with one ray of
your angle.
– 2.) Follow the base line out to zero, if you
are at 180 switch the protractor around.
– 3.) Trace to protractor up until you reach
the second ray of your angle.
– 4) The number your finger rests on is your
angle measure.
Part 4
Angle Relationships
Adjacent Angles
Adjacent Angles - are two angles that lie in
the same plane, have a common vertex, and
a common side, but no common interior
points
Complimentary Angles
Complementary Angles - Two angles whose
measures have a sum of 90
Complement Theorem
– If the non-common sides of two adjacent angles
form a right angle, then the angles are
complementary angles.
Angles complementary to the same angle or
to congruent angles are congruent
Supplementary Angles
Supplementary Angles - are two angles
whose measures have a sum of 180
Angles supplementary to the same
angle or to congruent angles are
congruent
Vertical Angles
Vertical Angles-are two non-adjacent angles formed
by two intersecting lines
Vertical Angles Theorem
– If two angles are vertical, then they are congruent
Linear Pair
Linear Pair - is a pair of adjacent angles who
are also supplementary
Supplement Theorem – If two angles form a linear pair, then they
are supplementary angles
Part 5
Perpendicular Lines and their
theorems
Perpendicular Lines
Lines that form right angles are perpendicular
– Perpendicular lines intersect to form 4 right angles
– Perpendicular lines form congruent adjacent
angles
– Segments and rays can be perpendicular to lines
or to other line segments or rays
– The right angle symbol in a figure indicates that
the lines are perpendicular.
Theorems
Theorem 2.9 - Perpendicular lines intersect to form
four right angles
Theorem 2.10 - All right angles are congruent
Theorem 2.11 - Perpendicular lines form congruent
adjacent angles
More Theorems
Theorem 2.12 - If two angles are
congruent and supplementary, the each
angle is a right angle
Theorem 2.13 - If two congruent angles
form a linear pair, then they are right
angles.
Unit 1
The End!