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INTRODUCTION TO GEOMETRY
UNIT 3: BASIC GEOMETRIC CONCEPTS
Undefined Terms Intro to Geometry
Building Blocks of Geometry There are four terms that we will use as building blocks in this course. Your textbook calls them “undefined terms”, as their meanings are generally accepted without formal definition.
This means that these terms are impossible to define without using words or phrases that need to be defined themselves!
It is very important that you understand these ideas, as we will use them to define new terms later in the course.
A set is a collection of objects such that it is possible to determine whether a given object belongs to the collection or not.
A point is a position in space and has no dimensions (length, width, or thickness). A point is represented by a dot and labeled with a capital letter. The point P is drawn and labeled below:
A line is an infinite set of points that continues in both directions forever. In this class we will assume that all lines are straight unless otherwise stated. There are two ways to denote a line:
1. By naming any two points on the line: AB or BA or AC etc.
P
A B C
- With a lowercase letter.
PRACTICE: Name the following line 3 different ways.
Question: Can you name the line Mr?
A plane is a set of points that form a flat surface extending indefinitely in all directions. Since a plane has no thickness, it is like a flat surface that extends infinitely along its length and width. We can denote a plane:
- By naming three points in the plane that are not on the same line: Plane ABC
- With a capital letter: Plane P
Example #1: Given the plane below answer the following questions.
a) Give another name for line n.
b) Give another name for plane C.
c) Give a set that contains point V.
m
A B
C
P
Q
R
S
T
V
n
k
C
P
K L M
r
Lines and Line Segments
Intro to Geometry
Picture Def: A collinear set of points is
Def: A noncollinear set of points is
Ex. 2: Given the figure illustrated below, answer the following questions:
a) Give two sets of collinear points.
b) Give two sets of noncollinear points.
Distances between Points
In the previous class we discussed the real number line, and stated that every point on the line has a coordinate that corresponds to a real number.
Given two points P and Q, we represent the distance from P to Q as PQ.
Ex. 3: a) Find AT, or the distance from point A to T. Is this the same as TA?
b) Find MT, HT, and HA.
A
W
R
C
X
M A T H
Definition of Betweenness: Given three collinear points, A, B, and C, B is between A and C if and
only if AB BC AC.
Ex. 4:
a) Using the definition of betweenness, show that C is between B and D.
b) Using the definition of betweenness, show that C is not between A and B.
Def: A line segment , or a segment, is
Using example 4 from above, we call the line segment bound by A and C AC.
Def: The length or measure of a line segment is
Def: Two segments are congruent if and only if
Symbol for congruent:
Ex****. 5 : The rectangle below consists of 4 line segments. Name the 4 segments and state which are congruent.
A B C D
A
B
F
E
C
D
REVIEW
- For each of the following, (a) find the value of x that makes the statement true and
(b) name the property that is illustrated in the equation.
a) 8 12 12 x b) x 6 0 c) 9 x 0
d) (^) 2(3 4) (3 4) x e) (^) x (2 4) 3(2) 3(4) d) (^4) x (^) 1
- Are there any real numbers that do not have additive inverses? Multiplicative inverses?
Midpoints, Segment Bisectors & Perpendicular Bisectors
Intro to Geometry
Definition: The midpoint of a line segment is a point of that line segment that divides the segment into two congruent segments.
If we are given two points with coordinates, we can find the midpoint by taking the average of the coordinates.
That is, for a line segment PQ , where P has coordinate a, and Q has coordinate b , mdpt PQ d ihas
coordinate
a b
Example #1: Give the coordinate of each midpoint, then sketch and label it on the number line.
a) mdpt CE d i M b) mdpt BD d i N c) mdpt AE d i K
Example #2: For the following questions, M is the midpoint of AB. Find x, AM and AB.
a) AM = 3x + 15, BM = 6x – 60 b) AM = 10, AB = 5x + 5
A M B A M B
A M B
M is the midpoint of because .
We can also write this as :
A B C D E
Midpoints, Segment Bisectors & Perpendicular Bisectors HOMEWORK
Intro to Geometry
1. If AM MB , does this necessarily mean that M is the midpoint of AB? Explain why or why not.
- S, M, and T are points on the number line. The coordinate of M is 2, and the coordinate of T is 14. What
is the coordinate of S if mdpt ST d i M?
3. P, Q, and R are points on the number line. The coordinate of P is 10 , the coordinate of Q is 6, and the
coordinate of R is 8.
a) Does PQ QR PR?
b) Is Q the midpoint of PR? If not, what is the coordinate of the midpoint?
4. In the diagram below CD and AB bisect each other at point K.
(a) If CK is five more than twice KB and AK 10 find CK.
(b) If CD is twice AB and AK 5 then find KD.
C
D
A K B
5. CD bisects AB at point E.
(a) Draw a diagram for this problem. If AB = 18, then find AE.
6. PQ bisects RS at point T. Which of the following equations is true?
(a) RT = PT (b) PT = TQ (c) RS = PQ (d) RT = TS
7. P is the midpoint of MN. If PN = x + 16 and MN = 4 x – 8, solve for the value of x. (draw a diagram)
Construction #2 – Construct the Midpoint of a Given Line Segment
Construction #3 – Construct the Perpendicular Bisector of a Given Line Segment
Both the midpoint and the perpendicular bisector can be drawn using the same construction.
1. Begin with line segment XY.
2. Place the compass at point X. Adjust the compass
radius so that it is more than
XY
. Draw two
arcs as shown here.
- Without changing the compass radius, place the
compass on point Y. Draw two arcs intersecting the
previously drawn arcs. Label the intersection points
A and B.
4. Using the straightedge, draw line AB.
(a) Label the intersection point M. Point M is
the midpoint of line segment XY.
(b) Line AB is perpendicular to line segment XY.
Given the line segment below, construct the perpendicular bisector of BR.
X Y
X Y
A
B
X Y
X Y
A
B
M
B R
Constructions – Class Practice
Intro to Geometry
1. Draw and label VW. Construct the perpendicular bisector of VW.
- Construct a segment that is (^4) CD .
3. Construct the bisector of segment KL
V
W
Turn the Page
AIM: To copy segments and construct the midpoint and perpendicular bisector of a given line segment using a compass and straightedge.
C D
K
L
Rays and Angles
Intro to Geometry
Definition : A ray is
A ray is labeled by its endpoint and one other point on the ray. Ray AB is written as AB.
Definition: An angle is
In this class we will measure angles in degrees. We will denote the degree measure of an angle ABC by m ABC.
We can also classify angles based on their degree measures.
Definition: An acute angle is an angle with a measure
Definition: A right angle is an angle with a measure
Definition : An obtuse angle is is an angle with a measure
Definition: A straight angle is an angle with a measure
Definition: A reflex angle is an angle with a measure
A B
We can write this angle in 3 ways:
- By its vertex:
- With three points (one on each side and the vertex in the middle): or.
- With a number labeled inside the angle:
A
B C
side
side
vertex^1
AIM: To review and define the terms RAY and Angle. To identify angles as acute, right, obtuse or straight.
To label and describe angles properly.
To define congruent angles and angle bisectors.
To be able to add and subtract angles (and segments).
Example 1: Use the figure to answer the following questions.
a) Name a right angle.
b) Name an obtuse angle.
c) Name two acute angles.
d) Name all FOUR rays in the diagram.
Example 2: Use the following figure to answer the questions below:
a) Is AB the same as BA? Why or why not?
b) Is AB the same as AD? Why or why not?
Def: Two angles are congruent if they have the same angle measure. We write A is
congruent to B as A B. We can also use numbered angles and angles named using 3 letters.
For example: 1 2 , ABC DEF. These can also be combined: 1 B , ABC C , CDE 2
Def: A bisector of an angle is a ray whose endpoint is the vertex of the angle, and that divides that
angle into two congruent angles.
3
1 2
A
B
C
D
E
A B C D
A
B
C
D
bisects because
Example 4: Use the figure to answer the following questions.
a) If m DBE 5 x 9 , and m EBC 4 x 18 ,
find x.
b.) If m ABC 135 , find m ABD.
Example 5: In the diagram below, it is given that BE bisects DBC and m DBE 72 .
(a) Find m DBC. (b) Find m DBA.
Example 6: In the following diagram, KM bisects LMN. If m LMK 10 7 and m KMN 8 x 9 , find x and
then find the measure of m LMN.
A
B
C
D
E
A
E
D
B C
N
M
K
L
Rays and Angles
Intro to Geometry HOMEWORK
Using the diagram to the right, give examples of each of the following.
1. Name a straight angle.
2. Name two acute angles.
3. Name two obtuse angles.
4. Name a right angle.
5. In each diagram, name each angle indicated in the picture.
a.) b.) c.) d.)
6. Classify each angle as acute , right , obtuse , or straight.
a.) ABC b.) 1 c.) FBC
d.) ABF e.) ABD f.) EBF
g.) ABE h.) FBA i.) DBF
7. In the following diagram, KM bisects LMN. If m ^ LMK^^ ^8 x ^14 and m ^ KMN^^ ^6 x ^10 , find x
and then find the measure of m LMN.
1 2
J
X M
K
L
P
Q
R
S
L
M (^) N
O A
D
C
B
E (^) E
F
G H
1 2
C
B F
D
E
A
N
M
K
L
