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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
Pairs of 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
Vertical Angles-are two non-adjacent angles
formed by two intersecting lines
Linear Pair - is a pair of adjacent angles who
are also supplementary
Angle Relationships
Complementary Angles - Two angles
whose measures have a sum of 90
Supplementary Angles - are two angles
whose measures have a sum of 180
Part 5
Angle Theorems
Theorem 2.3
Supplement Theorem – If two angles form a linear pair, then they
are supplementary angles
Theorem 2.4
Complement Theorem
– If the non-common sides of two adjacent
angles form a right angle, then the angles
are complementary angles.
Theorem 2.6
Angles supplementary to the same
angle or to congruent angles are
congruent
Theorem 2.7
Angles complementary to the same
angle or to congruent angles are
congruent
Theorem 2.8
Vertical Angles Theorem
– If two angles are vertical, then they are
congruent
Part 6
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!