# Chap 1 Notes

```Name: _________________________________Block:_____________Date: ____________________
Chapter 1
Essentials of
Geometry
Sections Covered:
1.1 Points, Lines, and Planes
1.2 Segments and Congruence
1.3 Midpoint and Distance Formulas
1.4 Measure and Classify Angles
1.5 Angle Pair Relationships
SOL G.3
The student will use pictorial representations, including computer software,
constructions, and coordinate methods, to solve problems involving symmetry and
transformation. This will include
a) investigating and using formulas for finding distance, midpoint, and slope;
Unit 2 Syllabus: Ch 1 Essentials of Geometry
Block Date Topic
7
9/22
1.1 Identify Points, Lines, and Planes
1.2 Use Segments and Congruence
Homework
Wkst: 1.1 Definitions and 1.2
8
9/24
1.3 Use Midpoint and Distance Formulas
Wkst: 1.3 Midpoint and Distance
9
9/26
Review Day
Wkst: 1.1-1.3 Review
10
9/30
Quiz 1.1-1.3
Spiral Review
11
10/2
1.4 Measure and Classify Angles
Wkst: 1.4 Angle Measurement
12
10/4
1.5 Describe Angle Pair Relationships
Wkst: 1.5 Angle Relationships
13
10/6
Review Day
Wkst: Chapter 1 Review
14
10/8
Chapter 1 Test
Spiral Review
***Syllabus subject to change due to weather, pep rallies, illness, etc
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Wednesday, and Thursday afternoons.
2
Notes 1.1 Essential Definitions
Definition
Diagram
Notation
1. Point
No dimension
A
2. Line
One dimension; extends
without end
l
S
R
3. Plane
Y
Two dimensions; extends
without end
X
Z
M
4. Line Segment
Part of a line that contains
two endpoints and all the
points between them
A
B
5. Ray
Part of line that consists
of the endpoint and all the
points on the line that
extend in one direction
Initial point
Beginning point of the ray
A
C
6. Opposite Rays
Rays with same initial
point and extend to form a
line
C
A
B
7. Collinear points
Points that lie on the same
line
X
Y
8. Coplanar points
Points that lie on the same
plane
Y
X
Z
M
Intersection:
 The intersection of two lines is _____________________.

The intersection of a line and a plane is ____________________.

The intersection of two planes is ________________________.
3
Let’s put these definitions to use and you will be able to see that these really are not that
difficult.
B
C
Name this line SEVEN different ways.
A
Using the same diagram to the right name

FOUR rays:

Opposite rays:

Line segments:
m
Using the diagram to the right:

Name the plane shaded in two different ways.

Identify at least FOUR pairs of collinear points.

Identify at a set of coplanar points.

Where do lines g and h intersect?

Where does line g intersect with plane F?
4
Postulate:
______________________________________________________________________
Theorem:
______________________________________________________________________
Segment Addition Postulate: If ________ is between ________ and ________, then
__________ + __________ = ___________.
Remember: “The ____________ is equal to the ___________________________.”
Example:
Congruent Segments: segments that have the same __________.
a) AB  PQ is read "segment AB is _________________ to segment PQ."
b) AB = PQ is read "AB equals PQ" or “The length of segment AB equals the length
of segment PQ.”
c) Since AB and PQ represent numbers, they are equal.
In contrast, AB and PQ represent figures and are called congruent.
Diagram:
Notation:
Example:
5
Example:
Example:
Together
Example: Point S is between R and T on RT . Us e the given information to write an equation in
terms of x. Solve the equation. Then, find RS and ST.
a. RS = 2x + 10
ST = x – 4
RT = 21
b. RS = 3x – 16
ST = 4x – 8
RT = 2x + 36
6
Notes 1.3: Midpoint & Distance
Midpoint: _______________________________________________________________________________
Segment bisector:_________________________________________________________________________
1.) Find AC if AB = 10 cm.
2.) Point W bisects UV .
Find UV if WV = 14 inches.
3.) Find FM.
4.) Point M is the midpoint of VW .
Find the length of VM .
PRACTICE: In the diagram, M is the midpoint of the segment. Find the indicated length.
a.
b.
7
Ex1: Given two endpoints can you FIND THE MIDPOINT?
The coordinates of a segment’s midpoint are
(______________, ________________)
The endpoints of AB are A(3, 2) and B(6, 7).
Find the coordinates of the midpoint M.
Find the coordinates of the midpoint of the segment with the given endpoints.
a. S (4, -1) and T (6, 0)
b. L (4, 2) and P (0, 2 )
c. H (-5, 5) and K (7, 3)
Ex2: Given an endpoint and a midpoint, can you find the other endpoint?.
Use the given endpoint R and midpoint M of RS to find the coordinates of the
endpoint S. R (1, 2) and M (-2, 1)
**Think…. DMSE
8
Ex3: Given two endpoints, can you find the distance?
Think… Pythagorean Thm
Find the length of the segment. Leave answer in simplest radical form and round to the nearest
tenth of a unit.
Ex4: Given two endpoints, can you evaluate to see if they are congruent?
9
Notes 1.4: Angle Relationships
1. An angle is a figure formed by 2 __________that have the same endpoint.
2. The rays of the angle are called __________ of the angle; the endpoint is called the ______________.
An angle is named three ways:
1) by 3 capital letters, but the vertex must be in the middle.
_______ or ________
A
2) by a number placed inside the angle.
_________
B
2
C
3) by the vertex...if there is only one angle with that vertex. Even though the angle is named four
different ways, it is the same angle.
_________
Adjacent angles are 2 angles that
1.
2.
3.
4.
6
4
1
2
________
3
________
5
________
8
7
_______
9
10
_______
Definition: Congruent angles are angles that have the same _______________.
Reminders…
1. Acute angle- angle with measure between _____ and _____.
2. Right angle- angle with measure of _______.
3. Obtuse angle- angle with measure between _____ and ______.
4. Straight angle- angle with measure of _________.
10
TWO BIG IDEAS (for angles):
1.
Angle Bisector Idea:
Ex. BT bisects ABC. Find the value of x.
Then find the measure of the angles to the nearest tenth.
C
T
(3X + 13)
B
(5X - 17)
A
Find x given the following:
A. m∠ABC = 98°
B. m∠ABC = 9x + 87°
11
2.
EX:  ADC = 50°. What does ADB and  BDC equal to the nearest tenth?
12
Notes 1.5: Angle Relationships
Vertical Angles:
Linear Pair:
Determine if the angles are vertical angles, linear pair, or neither.
a)
b)
c)
d)
e)
f) 5 and 6
g) 1 and 4
Classify the angles as linear pair, vertical angles or neither.
a)
2, 5
b)
4, 7
c)
8, 10
d)
3, 10
Find the value of each variable to the nearest tenth. Be sure to write an equation and circle the
1.
2.
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3.
4.
Complementary Angles:
Supplementary Angles:
1) Find the complement of 27.
2) Find the supplement of 115.
3) The following angles are complementary. Find the measure of the angles to the nearest tenth.
mA  5x  4,
mB  7x  10
4) The following angles are supplementary. Find the measure of the angles to the nearest tenth.
mA  11x  2,
mB  8x  7
5) Two angles form a linear pair. The measure of one angle is 8 times the measure of the other
angle. Find the measure of each angle to the nearest tenth.
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