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CHAPTER 1 – ESSENTIALS OF GEOMETRY
In this chapter we address three Big IDEAS:
1) Describing geometric figures
2) Measuring geometric figures
3) Understanding equality and congruence
Section:
1 – 1 Identify Points, Lines, and Planes
Essential
Question
How do you name geometric figures?
Key Vocab:
Undefined
Terms
A basic figure that is not defined in terms of other figures. The
undefined terms in geometry are point, line, and plane.
Point
An undefined term in geometry, it
names a location and has no size.
Line
An undefined term in geometry, a
line is a straight path that has no
thickness and extends forever.
”Point A”
A
S
T
“Line ST”; ST
E
Plane
F
An undefined term in geometry, it
is a flat surface that has no
thickness and extends forever.
D
G
Plane EFG ; Plane R
Collinear
Points
Coplanar
Points
Points that lie on the same line.
Points that lie in the same plane.
Line
Segment
Part of a line that consists of two
points, called endpoints, and all
points on the line that are between
the endpoints.
Ray
Part of a line that consists of a
point called an endpoint and all
points on the line that extend in
one direction.
Opposite
Rays
Collinear rays, with a common
endpoint, extending in opposite
directions.
C
B
BC
A
B
Ray AB; AB
R
S
T
SR and ST are opposite rays.
With point S as the common
Student Notes  Geom  Chapter 1 – Essentials of Geometry  KEY
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vertex.
Intersection The set of points the figure has in common.
Show:
Ex 1:
E
A
m
B
D
C
a. Give two other names for BD
DB and m
b.
Give another name for plane T.
plane ABE
c. Name three points that are collinear.
A, B, C
d. Name four points that are coplanar.
A, B, C, E
Ex 2:
a. Give another name for PR . RP
b. Name all ray with endpoint Q. Which of these rays are
opposite rays?
QP , QR , QT , QS; QT and QS , QP and QR are opposite rays.

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
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Section:
1 – 2 Use Segments and Congruence
Essential
Question
What are congruent segments?
Key Vocab:
Postulate
or Axiom
In Geometry, a rule that is accepted without proof
Between
When three points are collinear,
you can say one point is between
the other two.
A
B
C
Point B is between points A & C
A
B
C
D
Lengths are equal
Congruent
Segments
Line segments that have the same
length.
AB = CD (is equal to)
Segments are congruent
AB  CD (is congruent to)
Ruler Postulate:
The points on a line can be matched one to
one with the real numbers. The real
number that corresponds to a point is the
coordinate of the point.
The distance between points A and B,
written AB, is the absolute value of the
difference of the coordinates of A and B.
Student Notes  Geom  Chapter 1 – Essentials of Geometry  KEY
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Segment Addition Postulate:
AC
If B is between A and C, then
AB  BC  AC .
A
If AB  BC  AC , then B is between A
and C.
B
AB
C
BC
Show:
Ex 1: The cities shown on the map lie approximately in a straight line. Use the
given distances to find the distance from Bismarck to Fargo. 190 mi
Ex 2: Find CD. 25
Student Notes  Geom  Chapter 1 – Essentials of Geometry  KEY
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Ex 3: Graph the points X(-2, -5), Y(-2, 3), W(-4, 3), and Z(4, 3) in a coordinate plane. Are
XY and WZ congruent? Yes, both equal 8.
Student Notes  Geom  Chapter 1 – Essentials of Geometry  KEY
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Practice Level A
1. 16
2. 36
3. 21
4. 19
5. 11
6. 42
10.
congruent
11.
not congruent
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
5
4
6
9
15
10
9
9
37
28
Student Notes  Geom  Chapter 1 – Essentials of Geometry  KEY
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Section:
1 – 3 Use Midpoint and Distance Formulas
Essential
Question
How do you find the distance and the midpoint between two
points in the coordinate plane?
Key Vocab:
Midpoint
Segment
Bisector
The point that divides the
segment into two congruent
segments.
M is the midpoint of AB
A point, ray, line, line segment, or
plane that intersects the segment
at its midpoint.
CD is a segment bisector of AB
Key Concepts:
Midpoin
t
Formula
If A(x1, y1) and B(x2, y2)
are points on a coordinate
plane, then the midpoint M
of AB has coordinates
 x1  x2 y1  y2 
,


2 
 2
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Distanc
e
Formula
If A(x1, y1) and B(x2, y2)
are points in a coordinate
plane, then the distance
between A and B is
AB 
 x2  x1    y2  y1 
2
2
Show:
Ex 1: The figure shows a gate with diagonal braces. MO bisects NP at Q. If PQ=22.6 in.,
find PN. 45.2 in
Ex 2: Point S is the midpoint of RT . Find ST. 23
Ex 3:
a. The endpoints of GH are G(7, -2) and H(-5, -6). Find the coordinates of the
midpoint P. P(1, -4)
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Practice Level A
1. 12 cm
2. 34 cm
3. 54
1
2
in.
4. 28 ft
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
9
15
19
118
222
86
(3, 4)
(4, 5)
6.4
4; 0
Student Notes  Geom  Chapter 1 – Essentials of Geometry  KEY
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Section:
1 – 4 Measure and Classify Angles
Essential
Question
How do you identify whether an angle is acute, right, obtuse, or
straight?
Key Vocab:
Angle
Sides
Vertex
Congruent
Angles
Consists of two different rays
with the same endpoint
Ex: BAC , CAB, A, 1
The rays are the sides of the
angle
Ex: AB, AC
The common endpoint of the
rays
Ex: A
Angles that have the same
measure
A  B and m A  m B
Angle
Bisector
A ray that divides an angle into
two angles that are congruent
YW bisects XYZ
 XYW  WYZ
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Classifying Angles:
Acute
Angle
0  m A  90
Right Angle
m A  90
Obtuse
Angle
90  m A  180
Straight
Angle
m A  180
Postulate:
Angle
Addition
Postulate
If P is in the interior of RST ,
then the measure of RST is
equal to the sum of the measures
of RSP and PST.
m RST  m RSP  m PST
Show:
Ex 1: Name each angle that has N as a vertex.
__________
MNO, ONP, MNP 
__________
Student Notes  Geom  Chapter 1 – Essentials of Geometry  KEY
__________
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Ex 2: Use the diagram to find the measure of each angle and classify the angle.
C
B
A
E
a.
DEC
b.
DEA ___180o Straight___
c.
CEB
d.
DEB ____110o Obtuse___
___90o Right_______
____20o Acute_____
D
Ex 3: If m XYZ  72, find m XYW and m ZYW. 51o, 21o
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Practice Level A
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
 DEF,  FED, and  E; vertex: E; sides: E D and E
 JKL,  LKJ, and  K; vertex: K; sides: K J and K L
 QVS,  SVQ, and  V; vertex: V; sides:V and VQS
obtuse
acute
right
obtuse
30°; acute
50°; acute
105°; obtuse
180°; straight
123°
56°
64°
86°
23°
m ZXY = 39°, m WXY = 78°
m YXZ = 57°, m YXW = 114°
m WXZ = 73°, m ZXY = 73°
44°
Student Notes  Geom  Chapter 1 – Essentials of Geometry  KEY
F
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Section:
1 – 5 Describe Angle Pair Relationships
Essential
Question
How do you identify complementary and supplementary angles?
Key Vocab:
Complementary
Two angles whose sum is 900
Angles
Adjacent
Non-adjacent
m 1  m 2  90
Supplementary
Angles
Two angles whose sum is 1800
Adjacent
Non-adjacent
m 5  m 6  180
Adjacent
Angles
Two angles that share a
common vertex and side, but
have no common interior
points
Linear Pair
Two adjacent angles whose
noncommon sides are opposite
rays
Vertical Angles
<1 and <2 are a linear pair
Two angles whose sides form
two pairs of opposite rays
<1 and <3 are vertical angles
<2 and <4 are vertical angles
Student Notes  Geom  Chapter 1 – Essentials of Geometry  KEY
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Show:
Ex 1: In the figure, name a pair of complementary angles, a pair of supplementary angles,
and a pair of adjacent angles. GHI , JLK ; GHI, KLM; JLK, KLM
1 and
2 are complementary m 1  17, find m 2. 73o
b. Given that 3 and
4 are supplementary m 3  119, find m 4. 61o
Ex 2: a. Given that
Ex 3: Two roads intersect to form supplementary angles,
m XYW and m WYZ. 76o, 104o
XYW and WYZ. Find
Ex 4: Identify all of the linear pairs and all of the vertical angles in the figure.
2 and 3, 1 and 2; 1 and 3
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Ex 5: Two angles form a linear pair. The measure of one angle is 3 times the measure of
the other angle. Find the measure of each angle. 45o, 135o
Practice Level A
1. yes
2. no
3. no
4. complementary: POQ and ROS; supplementary: either POR and ROS or
POQ and QOS
5. complementary: UTV and XYZ; supplementary: VTW and XYZ
6. 38
7. 14
8. 71
9. 27
10. 33
11. 86
12. 142
13. 59
14. mABD = 36, mDBC = 54
15. mABD = 54, mDBC = 126
16. mABD = 101, mDBC = 79
17. linear pair
18. vertical angles
19. vertical angles
20. linear pair
21. linear pair
22. neither
23. neither
24. linear pair
25. x = 7, y = 17
26. x = 25, y = 6
27. x = 9, y = 4
28. mA = 60, mB = 30
29. mA = 44, mB = 46
30. mA = 18, mB = 72
31. mA = 33, mB = 57
32. mA = 45, mB = 135
33. mA = 158, mB = 22
Student Notes  Geom  Chapter 1 – Essentials of Geometry  KEY
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Section:
1 – 6 Classify Polygons
Essential
Question
How do you classify polygons?
Key Vocab:
Polygon
Sides
Vertex
Convex
A closed plane figure with three
or more sides, where each side
intersects exactly two sides, one
at each endpoint, so that no two
sides with a common endpoint
are collinear
Each line segment that forms a
polygon
Each endpoint of a side of a
polygon
Sides: AB, BC , CD, DE , and AE
Vertices: A, B, C, D and E
interior
A polygon where no line
containing a side of the polygon
contains a point in the interior of
the polygon
Convex Polygon
Concave
A polygon that is not convex
Concave Polygon
n-gon
A polygon with n sides
Equilateral
A polygon with all of its sides
congruent
A polygon with all of its interior
angles congruent
Equiangular
Regular
A convex polygon that has all
sides and all angles congruent
A polygon with 14 sides is a 14gon
Regular Pentagon
Student Notes  Geom  Chapter 1 – Essentials of Geometry  KEY
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Show:
Ex 1: Tell whether each figure is a polygon. If it is, tell whether it is concave or convex.
a.
b.
Yes; Convex
No; Concave
Ex 2: Classify the polygon by the number of sides. Tell whether the polygon is equilateral,
equiangular, or regular. Explain your reasoning.
a.
Triangle; only 2 sides congruent, only 2 anlges
congruent, so not equilateral, not equiangular, not
regular.
b.
Hexagon; equilateral, equiangular, regular
Ex 2 cont.: Classify the polygon by the number of sides. Tell whether the polygon is
equilateral, equiangular, or regular. Explain your reasoning.
c.
Quadrilateral; equilateral, not equiangular, so
not regular
Ex 3: A rack for billiard balls is shaped like an equilateral triangle. Find the length of a
side.
Student Notes  Geom  Chapter 1 – Essentials of Geometry  KEY
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Practice Level A
1. No; one side is not a segment.
2. Yes; concave
3. Yes; convex.
4. Quadrilateral; regular; the figure is equilateral and equiangular.
5. Hexagon; regular; the figure is equilateral and equiangular.
6. Pentagon; none; all sides are not equal and all angles are not equal.
7. Decagon; equilateral and equiangular; the figure is not convex so it is not regular.
8. 3
9. 8
10. 6
11. 11
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