Download pre-ap geometry unit 1.3-1.5 - cguhs

Lesson 1.3: Segments, Rays,
and Distance
Pre-AP Geometry
Points, Lines, and Planes

Line Segment

Two points (called the endpoints) and all the
points between them that are collinear with
those two points
Named line segment AB, AB, or BA
line AB
A
B
segment AB
A
B
Length of a segment



Length of BC is stated as BC. It is
the distance between points B and
C.
On a number line, length of a
segment is found by subtracting the
coordinates of the endpoints.
On a coordinate plane, length of a
segment is found using the distance
formula D = ( x2  x 1)2  ( y2  y1 )2
Examples

Find the length between 5 and -3
on the number line
Find the distance of segment AB if
A(-3, 5) and B(2, -7)
Postulates


Postulate: statement that is accepted
without proof
Segment Addition Postulate


If B is between A and C, then AB + BC =
AC
Ruler Postulate
The points on a line can be paired with the
real numbers in such a way that any two
points can have the coordinates 0 and 1
2. Once a coordinate system has been
chosen in this way, the distance between
any two points equals the absolute value
of the difference of their coordinates.
1.
E
Examples

EG = 7x + 3
EF = 3x + 8
FG = 2x + 1
Find
2. Find
3. Find
4. Find
1.
x:
EG:
EF:
FG:
F
G
Segment Length terms
Congruent- two objects that have the same size and
shape. We use the symbol ≅ to show that two objects are
congruent.
 Congruent segments- two segments with equal
lengths.
Example: DE ≅ FG
Midpoint of a segment: a point that divides a segment into
two congruent segments.
x2  x1 y2  y1
Midpoint formula: M = ( 2 , 2 )
Segment bisector: A line, segment, ray, or plane which
intersects a segment at its midpoint.

Points, Lines, and Planes

Ray


Initial Point


Part of a line that starts at a point and extends
infinitely in one direction.
Starting point for a ray.
Ray CD, or CD, is part of CD that contains
point C and all points on line CD that are
on the same side as of C as D

“It begins at C and goes through D and on
forever”
Points, Lines, and Planes

Opposite Rays


If C is between A and B, then CA and
CB are opposite rays.
Together they make a line.
A
C
B
Lesson 1.4: Angles
Parts of an angle


Sides of an angle are made up of
rays
The rays meet at a point called the
vertex
vertex
sides
Naming an angle

An angle can be named by the vertex, by the
3 points on the angle: the side, the vertex
and the other side, or a number inside the
angle.
G
1
I
H
The angle can be named ∠GHI, ∠IHG, ∠H, or ∠1
Classifying angles




Acute angle: Angle measuring
greater than 0° and less than 90°.
Obtuse angle: Angle measuring
greater than 90° and less than 180°
Right angle: An angle measuring
exactly 90°
Straight angle: An angle measuring
exactly 180°
Angle Postulates
Protractor Postulate:
On AB in a given plane, choose any point O between
A and B. Consider OA and OB and all the rays that
can be drawn from O on one side of AB. These rays
can be paired with real numbers from 0 to 180 in a
way such that:
a. OA is paired with 0, and OB with 180
b. If OP is paired with x, and OQ with y, the m∠POQ
= │x - y │
 Angle addition postulate:

-If B lies on the interior of ∠AOC, then m ∠AOB + m∠BOC
= m∠AOC
-If ∠AOC is a straight angle, then m∠AOB+m ∠BOC =
180.
Angle Vocabulary

Congruent Angles


Adjacent angles


Two angles with equal measures
Angles which share a vertex and a
common side, but no common interior
points
Angle bisector

A ray which divides an angle into two
congruent, adjacent angles
Congruence symbols and drawing
conclusions

Do not assume anything in
geometry. Just because two
segments look equal does not mean
that they are.
Postulates and Theorems Relating
Points,
Lines,
and Planes
Lesson
1.5
Pre-AP Geometry
Postulates
A point is defined by its location.
A line contains at least two points.
A plane contains at least three points not
all in one line.
Space contains at least four points not all
in one plane.
Postulates
Through any two points there is exactly
one line.
Through any three points there is at least
one plane and through any three noncollinear points there is exactly one plane.
If two points are in a plane, then the line
that contains the point is in that plane.
If two planes intersect, then their
intersection is a line.
Theorem
If two lines intersect, then they intersect
in exactly one point.
Theorem
Through a line and a point not in the line
there is exactly one plane.
Theorem
If two lines intersect, then exactly one
plane contains the lines.
Review Quiz – True or False
1.
A given triangle can lie in more than one
plane.
2.
Any two points are collinear.
3.
Two planes can intersect in only one
point.
4.
Two lines can intersect in two points.
Review Quiz – True or False
1.
A given triangle can lie in more than one
plane. False. Through a line and a point
not in the line there is exactly one plane.
2.
Any two points are collinear. True.
Two planes can intersect in only one
point. False. If two planes intersect,
then they intersection is a line.
Two lines can intersect in two points.
False. If two lines intersect, then they
intersect in exactly one point.
3.
4.
Problem Set 1.5
Written Exercises
p.25: # 1 – 13 (odd), 14 20