UNIT A NOTES – BASICS OF GEOMETRY
GEOMETRY
NAME: ___________________________________
DATE: ____________________ BLOCK: _______
Section #1  Geometric Notation & Symbols
I can…
A1 Identify basic geometric symbols
A2 Identify and describe collinear, coplanar points, and intersecting lines
A3 Correctly name geometric figures
 In geometric drawings, ________________________ tick marks indicate ___________________________
segments or angles (segments or angles that have equal measures).


→ “is congruent to” (same measure)
 In geometry, some words such as point, line, and plane are ____________________.
Figure
Description
Symbol
Example
Point
 Has no dimension
 Named using a ______________ ______________
NONE
A
Line
 Extends in ______ dimension
 Named using a single lowercase
______________ letter or by ________ points
______ the line
Plane
 Extends in _______ dimensions
 Named using a single ______________________
________________ letter or by ____________
points in the plane (Note: The three
points must be noncollinear)
A
Name
B C
l
M
B
NONE
D
A
C
 Collinear points are points that lie on the _____________ ___________.
 Coplanar points are points that lie on the _____________ _____________.
 Intersecting lines are lines that ______________ one another.
For #s 1−4, use the diagram to the right.
1)
C
Name a point that is collinear with A and B:
2)
Name a point that is noncollinear with A and B:
3)
Name a point that is coplanar with A, D, and E:
4)
Name a point that is noncoplanar with A, D, and E:
H
A
D
G
B
E
K
F
1
For #s 5–9, use the diagram to the right.
5)
Name three points that are collinear:
6)
Name four points that are collinear:
7)
Name a point that is noncollinear with J and H:
8)
Name a point that is coplanar with L, P, and R:
9)
Name a line that is not intersecting KJ :
L
N
K
R
P
J
Figure
Description
Segment
 _________ of a line consisting of two ______________
and all the points ________________ them
 Named using the two ________________
Ray
M
Q
H
Symbol
Example
A
 Starts at an __________________ and extends
without end in one _______________________
 Named using the endpoint ____________, then
another point _______ the ray
B
A B
Name
C D
C
 Opposite rays are two rays that:
 Are ______________________.
 Extend in _________________________ directions.
 Have a _____________________ endpoint.
For #s 10–13, use the figure to the right.
10)
Does ADB name a line segment in the diagram? Why or why not?
11)
Does DB contain point A? Why or why not?
12)
Does AD contain point G? Why or why not?
13)
Are BA and DG opposite rays? Why or why not?
A
D
B
G
 An angle consists of two different _________ that have the same endpoint.
 The rays are the ___________ of the angle.
 The endpoint is the ______________ of the angle.
2
For #s 14–15, name the angle that is in bold.
14)
15)
R
Method
Description
C
8
A
1. Use the _____________ and
a point from each ________.
2. Use the _____________ only.
3. Use a ___________________.
D
T
3
4
Q
S
E
The vertex letter is
always in the _____________.
Can be used if there is
only ________ angle at the
vertex.
Can be used if there is
only ________ angle at the
vertex.
Section #2  Segments and Their Measures (Part 1)
I can…
A4 Describe the difference between a theorem and a postulate
A5 Use the Segment Addition Postulate to find the measure of a segment
 When three points lie on a line, you can say that one of them is between the other two.
A
B
C
Example: Point B is between points A and C.
 AB represents the ______________ of AB and the ________________________ between points A and B.
 In geometry, postulates (or axioms) are rules that are accepted _________________ _____________.
 A theorem, is a rule that must be ________________ true.
Segment Addition Postulate
AC
If B is between A and C, then ________ + ________ = ________
(And, if AB + BC = AC, then B is between A and C.)
16)
A
B
AB
BC
C
Michael Scofield is driving on Route 40 from
Little Rock to Nashville to find his true love
Sara. He stops in Memphis for lunch. The
distance from Little Rock to Memphis is 139
miles, and the distance from Little Rock to
Nashville is 359 miles. How far does Michael
need to travel after lunch to reach Nashville?
3
For #s 17–18, point J is between H and K.
17)
If HJ = 2x + 5, JK = 3x – 7, and KH = 18, find the value of x, HJ, and JK.
18)
If HJ = 10x + 8, HK = 12x + 4, and JK = 20, find the value of x and the length of the entire
segment.
In the diagram below, FK = 24, FG = 6, and HK = 14. Find GH.
You Try!
#19
K
G
H
F
20)
Aaron Harrison stopped by Graeter’s to get a double scoop on his way to Louisville for the big
game. The cone is 5 inches tall, the first scoop is 25.5–3x inches tall, followed by the final
scoop which is 2x – 13 inches tall. The whole cone (ice cream and all) stands 10 inches tall.
a. Find the height of the top scoop (in inches).
b. The distance from Lexington to Louisville is about 78 miles. If you lined the waffle cones
up back to back, how many would Harrison have to eat to reach the KFC Yum! Center in
Louisville? (Note: 1 mi = 5280 ft)
Go
Wildcats!
Cardinals
rule!
4
Section #3  Segments and Their Measures (Part 2)
I can…
A6 Accurately measure a segment with a ruler
A7 Use the Distance Formula to find the length of a line segment
A8 Use the Midpoint Formula to find the midpoint of a line segment
 Remember, AB represents the length of AB and the distance between points A and B.
21)
Find AB and CD to the nearest tenth of a centimeter.
A
B
C
D
The Distance Formula
B(x2, y2)
If A(x1, y1) and B(x2, y2) are points in a coordinate
plane, then the distance between A and B is
y2 y1
AB  (x2  x1 )2  ( y2  y1 )2
22)
A(x1, y1)
x2 x1
C(x2, y1)
In the graph below, is AB  CD ?
You Try!
#23
Find the length of EF given the coordinates below.
E(2, 3) and F(5, 1)
5
 The midpoint of a segment is the point that _________________, or bisects the segment into two
congruent segments. (In other words, it _________ ______ ______ __________.)
 If you know the coordinates of the endpoints of a segment, you can calculate the coordinates of
the midpoint.
The Midpoint Formula
If A(x1, y1) and B(x2, y2) are points in a coordinate
plane, then the midpoint of AB has coordinates
B(x2, y2)
y2
y1 + y2
 x1  x 2 y1  y2 
,


2 
 2
(
2
y1
A(x1, y1)
x1
x1 + x2 y1 + y2
,
2
2
x1 + x2
)
x2
2
24)
QS has endpoints Q(3, 5) and S(7, 9). Find the coordinates of its midpoint.
25)
Find the midpoint of the segment graphed below.
If L(7, 8) and M(1, 6), find the coordinates of the midpoint of LM .
You Try!
#26
6
27)
Given the midpoint, M, of AB and one endpoint, find the coordinates of the other endpoint.
a. M(3, 9) and A(3, 2)
b. M(4, 10) and B(10, 9)
Section #4  Angles and Their Measures
I can…
A9 Accurately measure an angle with a protractor
A10 Use the Angle Addition Postulate to find the measure of an angle
 “mA” means “the ___________________________ of angle A.”
 Angles are measured in ____________________.
28)
Name each angle and find its measure the nearest degree.
a. Name:
____________
Measure: ____________
b. Name:
____________
Measure: ____________
M
X
L
Z
N
Y
 Angles are classified as acute, right, obtuse, and straight, according to their ______________________.
A
_____________________
measure is between
_______ and _______
A
_____________________
measure equals
_______
A
_____________________
measure is between
_______ and _______
A
_____________________
measure equals
_______
7
 Two angles are adjacent angles if they share a common ________________ and ___________, but have
no common interior points.
29)
Sketch two adjacent angles so that one angle is acute and one angle is obtuse.
Angle Addition Postulate
R
If P is in the interior of RST, then
____________ + ____________ = ____________
S
P
1
2
T
30)
If mATM = (5x + 21)°, mMTH = (7x + 3)°, and mATH = 18x, find the value of x.
A
M
T
31)
H
If mAED = 160° and mCEB = 110°, find mDEB.
D
E
C
A
B
If mABE = 24°, mDBC = 46°, and mDBE = 88°, find mABC.
You Try!
#32
D
C
E
A
B
8
33)
The graph below shows points of sail (sailing positions.)
a. Suppose a sailboat is in the running position. How many degrees must the sailboat be
turned so that it is in the close reach position?
b. The expression 12x + 6 represents the number of degrees between the close hauled and
beam reach positions. What is the value of x?
Section #5  Proving Statements About Angles
I can…
A11 Logically order the steps in a proof
 Remember, a theorem is a statement that follows as a result of other proven statements and a
postulate is a statement that is assumed to be true (without proof).
 Given statements are statements that you may __________________ to be true. These form the
___________________ points for your proof.
 A two–column proof has _________________________ statements and _____________________ that show the
_________________ order of an argument.
 In Geometry, a theorem must be proven true before we can use it to solve problems.
 When writing a proof you cannot ____________________ anything to be true unless you have evidence
it is true. For instance, you cannot say that 1 below is a right angle just because it looks like it
is. In order to say it is a right angle you must ______________ it to be true or it must be provided to
you as a ______________ statement.
 Things you can use as reasons to justify your answer are ______________ statements,
___________________________, _________________________, and ___________________.
9
34)
Given: 1 is a right angle
2 is a right angle
Prove: 1  2
2
1
Statements
Reasons
 You just proved the theorem that says all right angles are congruent!
Section #6  Angle Bisectors
I can…
A12 Solve problems involving angle bisectors
 An angle bisector divides an angle into two _______________________ angles.
For #s 3536, RQ bisects PRS.
35)
Find the mPRS.
P
(x+40)
Q
(3x-20)
R
36)
S
If PRS = 130°, find the value of x.
P
Q
(5x-15)
R
S
10
If WY bisects XWZ and mXWZ is 70°, find mXWY.
You Try!
#37
X
Y
W
Z
Section #7  Angle Pair Relationships
I can…
A13 Identify and sketch examples of vertical angles and linear pairs
A14 Solve problems involving the measures of vertical angles and linear pairs
A15 Solve problems involving complementary and supplementary angles
Relationship
Vertical Angles
Linear Pair
Description/Definition
Figure
Examples
 Two nonadjacent angles formed by a pair of
_____________________________ ___________.
 Vertical angles are _______________________.
 Two adjacent angles whose non-common
sides are _______________________ __________.
 The sum of the measures of angles that form a
linear pair is _________.
For #s 38–39, use the figure to the right.
38)
Name one pair of vertical angles.
5
1
39)
Name two linear pairs.
40)
Using the diagram below, find the value of x.
(6x-15)°
2
4
3
105°
(4x+15)°
11
41)
Using the diagram below, find the value of y.
(5y+10)°
(3y-4)°
Find the values of x and y in the diagram below.
You Try!
#42
120°
x°
y°
43)
Using the diagram below, find the values of x and y.
(9x 22)°
(7x+12)°
(7y)°
Relationship
Description/Definition
Complementary
Angles
 Two angles are complementary angles if the sum
of their measures is ________.
 Each angle is the _________________________ of the other.
 Can be adjacent or non adjacent.
Supplementary
Angles
 Two angles are supplementary angles if the sum
of their measures is ________.
 Each angle is the __________________________ of the
other.
 Can be adjacent or non adjacent.
Figure
12
44)
Find the complement and supplement of a 47°angle.
Complement: ______________________________
Supplement: ______________________________
45)
Find the complement and supplement of a 110°angle.
Complement: ______________________________
Supplement: ______________________________
Find the complement and the supplement of a 36° angle.
You Try!
#46
47)
One of two supplementary angles is twice the other. Find the measure of both angles.
13
UNIT A LEARNING TARGETS (GOALS)
Date Goal
I can…
#s
A1
Identify basic geometric
symbols
#115
A2
Identify and describe
collinear, coplanar points,
and intersecting lines
#19
A3
Correctly name geometric
figures
#115
A4
A5
A6
A7
A8
Describe the difference
between a theorem and a
postulate
Use the Segment Addition
Postulate to find the
measure of a segment
Accurately measure a
segment with a ruler
Use the Distance Formula to
find the length of a line
segment
Use the Midpoint Formula
to find the midpoint of a line
segment
#19
#21
#23
#26
A9
Accurately measure an
angle with a protractor
#28
A10
Use the Angle Addition
Postulate to find the
measure of an angle
#32
A11
Logically order the steps in
a proof
#34
A12
Solve problems involving
angle bisectors
#37
A13
A14
A15
Identify and sketch
examples of vertical angles
and linear pairs
Solve problems involving
the measures of vertical
angles and linear pairs
Solve problems involving
complementary and
supplementary angles
#38–39
#42
#46
Got it!
Self Evaluation
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14