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Chapter 1 Essentials of Geometry
Chapter 1
Essentials of Geometry
1.1 Identify Points, Lines, and Planes
Objective:
Name and sketch geometric figures so you can use geometry terms in the real world.
Essential Question: How do you name geometric figures?
Common Core: CC.9-12.G.CO.1
Vocabulary:
Point: a point has no dimension. It is represented by a dot.
Line: A line has one dimension. Through any two points, there is exactly one line.
A
A
B
l
Plane: a plane has two dimensions. It is represented by a shape that looks like piece of paper, but extends without end.
Through any 3 points not on the same line, there is exactly one plane. Use three points to name a plane.
A
B
C
M
EXAMPLE 1 Name points, lines, and planes
b. Name three points that are collinear.
c. Name four points that are coplanar.
a. Give two other names for PQ and for plane R.
• A line segment is part of a line containing two endpoints
and all points between them.
•A line segment is named using
its endpoints.
•The line segment is named
segment AB or segment BA
•The symbol for segment AB is
A B
Rays and line segments are parts of lines.
A ___________ has a definite beginning and end.
AB
A ___ has a definite starting point and extends
without end in one direction.
RAY: The starting point of a ray is called the ________.
A ray is named using the endpoint first, then another
point on the ray.
The ray above is named ray AB.
A B
Rays and line segments are parts of lines.
The symbol for ray AB is AB
___________ are two rays that are part of a the
same line and have only their endpoints in common.
X Y Z
XY and XZ are ____________.
The figure formed by opposite rays is also referred
to as a ____________.
1) Name two segments.
A
C
B
U
D Possible Answers:
2) Name a ray. Possible Answers:
Intersecting planes practice:
Homework: 1.1 Exercises
Concepts: #1-11, 12 - 38 even, 40 – 43
Regular: # 1 – 11, 12 - 20 even, 21 - 43
Honors: # 1 – 44
1.2 Use Segments and Congruence
Objective: Use segment postulates to identify congruent segments
Essential Question: What are congruent segments?
Common Core: CC.9-12.G.CO.1
CC.9-12.G.CO.7
Postulate:
A rule that is accepted without proof.
aka: axiom
The distance between two points A and B on a number line can be found by using the Ruler Postulate.
Ruler
Postulate
The points of a line can be put into one-to-one correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers.
measure = a – b
B A
a b
Congruent Statements
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In geometry, two segments with the same length
are called ________ _________
they have the same length
Definition
of
Congruent
Segments
Two segments are congruent if
and only if
____________________
B A C
R
In the figures at the right, AB is
congruent to BC, and PQ is
congruent to RS.
The symbol is used to
represent congruence.
AB BC, and PQ RS.
Measuring Segments and Angles
x
11 -5 -3 -1 1 3 5 7 9 -6 -4 0 4 8 2 10 6 -2
R S T Y
Use the number line to determine if the statement
is True or False. Explain you reasoning.
RS TY
So, RS is not congruent to TY,
and the statement is false.
Because RS = 4 and TY = 5, TYRS
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Segment Addition Postulate
If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC
Use the Segment Addition
Postulate to write an equation.
AN + NB = AB Segment Addition Postulate
If AB = 25, find the value of x. Then find AN and NB.
Homework: 1.2 Exercises
Concepts: 1 – 30, 32 – 34
Regular: #1 – 30, 32 - 35
Honors: #1 – 36
*** skip #3-#5 (measuring)
1.3 Use Midpoint and Distance Formulas
Objective:
Find the lengths of segments in the coordinate plane
Essential Question:
How do you find the midpoint and distance of two points on the coordinate plane?
Common Core: CC.9-12.G.GPE.7
You will learn to find the coordinates of the midpoint of a segment.
bisects
The midpoint of a line segment, , is the point C that ______ the segment.
AB
B A C
-7 -6 -5 -4 -2 -3 0 -1 1 2 3 4 5 6 7
A B C
C = [3 + (-5)] ÷ 2
= (-2) ÷ 2
= -1
Use the definition of midpoint to write an equation.
RM = MT Definition of midpoint
M is the midpoint of RT.
Find RM, MT, and RT.
0
y
0 x
10 -1 2 4 6 8 10
10
-1
2
4
6
8
10
-2 3 7 -2
1
5
9
1 9
3
-2 -2
5
7
Find the midpoint, C(x, y), of a segment on the coordinate plane.
Consider the x-coordinate:
x = 1 x = 9
y = 7
y = 3
Consider the y-coordinate:
A
B
x
y C(x, y)
The Midpoint Formula
The coordinates of the midpoint M of
with endpoints A(x1, y1) and
B(x2, y2) are M
1 2 1 2,2 2
x x y y
O
y
x
1 1( , )x y
2 2( , )x y
1 2 1 2,2 2
x x y y
AB
x-coordinate of B
0
y
0 x
10 -1 2 4 6 8 10
10
-1
2
4
6
8
10
-2 3 7 -2
1
5
9
1 9
3
-2 -2
5
7
y-coordinate of B
Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B.
A(7, 2)
B(x, y) is somewhere over
there.
B(-1, 8)
midpoint
C(3, 5)
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The Distance Formula
The distance d between two points A (x1, y1) and B (x2, y2) is
212
2
12 yyxxd
Example: Find the distance between R (-2, 6) and S (6, -2) to the nearest tenth. Note: be careful using your calculator – use at
the very end.
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Homework: 1.3 Exercises
Concepts: #1, 2 – 24 even, 25 – 39 odd, 48, 49
Regular: #1, 2 – 24 even, 25 – 39 odd, 48 - 52
Honors: #1, 2 – 22 even, 23, 24 – 42 even, 48 - 52
1.4 Measure and Classify Angles
Objective:
Name, measure, and classify angles to identify congruent angles.
Essential Question:
How do you identify if an angle is acute, obtuse, right, or straight?
Common Core: CC.9-12.G.CO.1
CC.9-12.G.CO.7
There is another case where two rays can have a common endpoint.
R
S
T
This figure is called an _____.
Some parts of angles have special names.
The common endpoint is called the ______,
vertex and the two rays that make up the sides of the angle are called the sides of the angle.
side
R
S
T
vertex
side
There are several ways to name this angle.
1) Use the vertex and a point from each side. or
The vertex letter is always in the middle.
2) Use the vertex only.
If there is only one angle at a vertex,
then the angle can be named with
that vertex.
3) Use a number.
1
Definition of Angle
An angle is a figure formed by two noncollinear rays that have a common endpoint.
E
D
F
2
Symbols: DEF
2
E
FED
Naming Angles (Video)
B
A
1
C
1) Name the angle in four ways.
2) Identify the vertex and sides of this angle.
vertex:
sides:
Protractor Postulate
For every angle, there is a unique positive number between __ and ____ called the degree measure of the angle.
B
A
C
n°
0
180
m ABC = n
and 0 < n < 180
You can use a _________ to measure angles and sketch angles of given measure.
protractor
Q
R S
1) Place the center point of the protractor on vertex R. Align the straightedge with side RS.
2) Use the scale that begins with 0 at RS. Read where the other side of the angle, RQ, crosses this scale.
J
H
G
S Q R
m SRQ =
Find the measurement of:
m SRJ =
m SRG =
m QRG =
m GRJ =
m SRH
Once the measure of an angle is known, the angle can be classified as one of three types of angles. These types are defined in relation to a right angle.
Types of Angles
A
right angle
m A = 90
acute angle
0 < m A < 90
A
obtuse angle
90 < m A < 180
A
Classify each angle as acute, obtuse, or right.
110°
90° 40°
50°
130° 75°
Obtuse
Obtuse
Acute
Acute Acute
Right
5x - 7
B
The measure of B is 138. Solve for x.
9y + 4 H
The measure of H is 67. Solve for y.
B = 5x – 7 and B = 138
Given: (What do you know?)
H = 9y + 4 and H = 67
Given: (What do you know?)
Angle Addition Postulate
For any angle PQR, if A is in the interior of PQR, then mPQA + mAQR = mPQR.
2
1
A
R
P
Q
m1 + m2 = mPQR.
There are two equations that can be derived using this postulate.
m1 = mPQR – m2
m2 = mPQR – m1
These equations are true no
matter where A is located in the interior of PQR.
2
1 Y
Z
X
W
Find m2 if mXYZ = 86 and m1 = 22.
2x°
(5x – 6)°
B
D
C
A
Find mABC and mCBD if mABD = 120.
mABC + mCBD = mABD
Congruent Angles
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Angles with the same measure are _________________________
21 mmIf , then 21
Angle Bisector: a ray that divides an angle into
two congruent angles
Homework: 1.4 Exercises
Concepts: #1, 2, 3– 39odd, 51
Regular: #1, 2 – 48 even, 51, 52
Honors: #1, 2 – 50 even, 51, 52
1.5 Describe Angle Pair Relationships
Objective:
Use special angle relationships to find angle measures
Essential Question:
How do you identify complementary and supplementary angles?
Common Core: CC.9-12.G.CO.1
CC.9-12.G.CO.9
Definition of
Congruent
Angles
Two angles are congruent iff, they have the same ______________.
50° B
50°
V
B V iff
mB = mV
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1 2
To show that 1 is congruent to 2, we use ____.
Z X
To show that there is a second set of congruent angles, X and Z, we use double arcs.
X Z
mX = mZ
This “arc” notation states that:
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Definition of
Complementary
Angles 30°
A
B C
60° D
E
F
Two angles are complementary if and only if (iff) the sum of their degree measure is 90.
mABC + mDEF = 30 + 60 = 90
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Definition of
Supplementary
Angles
If the sum of the measure of two angles is 180, they form a special pair of angles called supplementary angles.
Two angles are supplementary if and only if (iff) the sum of their degree measure is 180.
50°
A B
C
130°
D
E F
mABC + mDEF = 50 + 130 = 180
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Definition of
Adjacent
Angles
Adjacent angles are angles that:
J
N
R 1
2
1 and 2 are adjacent
with the same vertex R
and common side RM
A) share a common side
B) have the same vertex, and
C) have no interior points in common
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When two lines intersect, ____ angles are formed.
1 2
3 4
There are two pair of nonadjacent angles.
These pairs are called _____________.
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Definition of
Vertical
Angles
Two angles are vertical iff there are two nonadjacent angles formed by a pair of intersecting lines.
1 2
3 4
Vertical angles:
1 and 3
2 and 4
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Vertical Angle
Theorem
Vertical angles are congruent.
1
4
3
2 m n
1 3
2 4
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Definition of
Linear Pairs
Two angles form a linear pair if and only if (iff):
1 and 2 are a linear pair.
A) they are adjacent and
B) their noncommon sides are opposite rays
A D B
1 2
AD form and BDBA
180 2 1
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1) If m1 = 4x + 3 and the
m3 = 2x + 11, then find the m3 1 2
3 4
2) If m2 = x + 9 and the m3 = 2x + 3,
then find the m4
3) If m2 = 6x - 1 and the m4 = 4x + 17,
then find the m3
4) If m1 = 9x - 7 and the m3 = 6x + 23, then
find the m4
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Find the value of x in the figure:
The angles are vertical angles.
So, the value of x is 130°. 130°
x°
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Find the value of x in the figure:
The angles are vertical angles.
(x – 10) = 125. (x – 10)°
125°
x – 10 = 125.
x = 135.
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(x – 10)°
Suppose two angles are congruent. What do you think is true about their complements?
1 2
1 + x = 90 2 + y = 90
x = 90 - 1 y = 90 - 2
x = y
x = 90 - 1 y = 90 - 1
Because 1 2, a “substitution” is made.
x is the complement of 1
y is the complement of 2
If two angles are congruent, their complements are congruent.
x y
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Homework: 1.5 Exercises
Concepts: #1, 2 – 52 even
Regular: #1, 2 – 52 even
Honors: #1, 2 – 52 even
1.6 Classify Polygons
Objective:
Classify polygons
Essential Question:
How do you classify polygons?
Common Core: CC.9-12.G.GMD.4
CC.9-12.G.MG.1
A polygon is a _____________ in a plane formed by segments, called sides.
A polygon is named by the number of its _____ or ______.
A triangle is a polygon with three sides. The prefix ___ means three.
Prefixes are also used to name other polygons.
Prefix Number of
Sides
Name of
Polygon
tri-
quadri-
penta-
hexa-
hepta-
octa-
nona-
deca-
3
4
5
6
7
8
9
10
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
U
T S
Q
R
P
A vertex is the point of intersection of two sides.
A segment whose endpoints are nonconsecutive vertices is a diagonal.
Consecutive vertices are the two endpoints of any side.
Sides that share a vertex are called consecutive sides.
An equilateral polygon has all _____ congruent.
An equiangular polygon has all ______ congruent.
A regular polygon is both ___________ and ___________.
equilateral but not
equiangular
equiangular but not
equilateral
regular, both equilateral and equiangular
Investigation: As the number of sides of a series of regular polygons increases,
what do you notice about the shape of the polygons?
A polygon can also be classified as convex or concave.
If all of the diagonals lie in the interior of the figure, then the polygon is ______.
If any part of a diagonal lies outside of the figure, then the polygon is _______.
Example: Classify the polygon shown at the right by the number of sides. Explain how you know that the sides of the polygon are congruent and that the angles of the polygon are congruent.
First, write and solve an equation to find the value of x. Use the fact that the sides of a regular hexagon are congruent.
Example: A table is shaped like a regular hexagon. The expressions shown represent side lengths of the hexagonal table. Find the length of a side.
Homework: 1.6 Exercises
Concepts: #1 – 30, 32 – 36
Regular: #1 – 30, 32 - 36
Honors: # 1 – 30, 32 – 36, 39, 40
Chapter 1 Test
