Essentials of
Geometry
Eleanor Roosevelt High School
Geometry
Mr. Chin-Sung Lin
ERHS Math Geometry
Basic Definitions
Mr. Chin-Sung Lin
ERHS Math Geometry
Definition
A definition is a statement of the precise
meaning of a term
A good definition must be expressed in words
that have already been defined or in words
that have been accepted as undefined
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulate
A postulate is an accepted statement of fact
Mr. Chin-Sung Lin
ERHS Math Geometry
Undefined Terms:
Set, Point, Line & Plane
Mr. Chin-Sung Lin
ERHS Math Geometry
Undefined Terms
Set
A collection of objects such that it is possible
to determine whether a given object belongs
to the collection or not
Mr. Chin-Sung Lin
ERHS Math Geometry
Undefined Terms
Point
A
B
C
D
E
A point indicates place or location and has no
size or dimensions
A point is represented by a dot and named by
a capital letter
Mr. Chin-Sung Lin
ERHS Math Geometry
Undefined Terms
A
B
Line
A line is a set of continuous points that form a
straight path that extends without ending in
two opposite directions
A line has no width
Mr. Chin-Sung Lin
ERHS Math Geometry
Undefined Terms
A
B
Line
A line is identified by naming two points on
the line. The notation AB is read as “line AB”
Points that lie on the same line are collinear
Mr. Chin-Sung Lin
ERHS Math Geometry
Undefined Terms
Plane
R
A plane is a set of points that form a flat
surface that has no thickness and extends
without ending in all directions
A plane is represented by a “window pane”
Mr. Chin-Sung Lin
ERHS Math Geometry
Undefined Terms
B
Plane
C
A
A plane is named by writing a capital letter in
one of its corners or by naming at least three
non-colinear points in the plane
Points and lines in the same plane are
coplanar
Mr. Chin-Sung Lin
ERHS Math Geometry
Undefined Terms
A
B
Postulate
Through any two points there is exactly one
line
Mr. Chin-Sung Lin
ERHS Math Geometry
Undefined Terms
P
Postulate
If two lines intersect, then they intersect in
exactly one point
Mr. Chin-Sung Lin
ERHS Math Geometry
Undefined Terms
Postulate
If two planes intersect, then they intersect in
exactly a line
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of
Real Numbers
Mr. Chin-Sung Lin
ERHS Math Geometry
Addition & Multiplication
Operation Properties
 Closure
 Commutative Property
 Associative Property
 Identity Property
 Inverse Property
 Distributive Property
 Multiplication Property of Zero
Mr. Chin-Sung Lin
ERHS Math Geometry
Closure
 Closure property of addition
The sum of two real numbers is a real number
a+b
is a real number
 Closure property of multiplication
The product of two real numbers is a real number
ab
is a real number
Mr. Chin-Sung Lin
ERHS Math Geometry
Commutative Property
 Commutative property of addition
Change the order of addition without changing the sum
a+b=b+a
 Commutative property of multiplication
Change the order of multiplication without changing the
product
ab =ba
Mr. Chin-Sung Lin
ERHS Math Geometry
Associative Property
 Associative property of addition
When three numbers are added, the sum does not
depend on which two numbers are added first
(a + b) + c = a + (b + c)
 Associative property of multiplication
When three numbers are multiplied, the product does not
depend on which two numbers are multiplied first
(a  b)  c = a  (b  c)
Mr. Chin-Sung Lin
ERHS Math Geometry
Identity Property
 Additive identity
When 0 is added to any real number a, the sum is a
a+0=a
and 0 + a = a
 Multiplicative identity
When 1 is multiplied to any real number a, the product
is a
a1=a
and 1  a = a
Mr. Chin-Sung Lin
ERHS Math Geometry
Inverse Property
 Additive inverses
Two real numbers are additive inverses, if their sum is 0
a + (-a) = 0
 Multiplicative inverses
Two real numbers are multiplicative inverses, if their
product is 1
a  (1/a) = 1 (for all a ≠ 0)
Mr. Chin-Sung Lin
ERHS Math Geometry
Distributive Property
 Multiplication distributes over addition
a  (b + c) = a  b + a  c
(a + b)  c = a  c + b  c
Mr. Chin-Sung Lin
ERHS Math Geometry
Multiplication Property of Zero
 Zero has no multiplicative inverse
 Zero product property
ab=0
if and only if
a = 0 or b = 0
Mr. Chin-Sung Lin
ERHS Math Geometry
Exercise
Identify the additive and multiplicative inverses of the
following nonzero real numbers:
 9
 -6
 d
 -b
 (3 – b)
Mr. Chin-Sung Lin
ERHS Math Geometry
Exercise
Identify the additive and multiplicative inverses of the
following nonzero real numbers:
 9
-9
 -6
6
 d
-d
 -b
b
 (3 – b)
(b – 3)
Mr. Chin-Sung Lin
ERHS Math Geometry
Exercise
Identify the additive and multiplicative inverses of the
following nonzero real numbers:
 9
-9
1/9
 -6
6
-1/6
 d
-d
1/d
 -b
b
-1/b
 (3 – b)
(b – 3)
1/(3-b)
Mr. Chin-Sung Lin
ERHS Math Geometry
Exercise
Identify the properties in the following operations:
 6  (1/6) = 1
 7 + (4 + a) = (7 + 4) + a
 34=43
 7  (x + 2) = 7  x + 7  2
 12 + 0 = 12
Mr. Chin-Sung Lin
ERHS Math Geometry
Exercise
Identify the properties in the following operations:
 6  (1/6) = 1
(multiplicative inverses)
 7 + (4 + a) = (7 + 4) + a
(associative)
 34=43
(commutative)
 7  (x + 2) = 7  x + 7  2
(distributive)
 12 + 0 = 12
(additive identity)
Mr. Chin-Sung Lin
ERHS Math Geometry
Lines &
Line Segments
Mr. Chin-Sung Lin
ERHS Math Geometry
Distance between Tow Points
Distance
A
B
a
b
The distance between two points on the real
number line is the absolute value of the
difference of the coordinates of the two points
AB =| a – b | = | b – a |
Mr. Chin-Sung Lin
ERHS Math Geometry
Order of Points
Betweenness
A
B
C
a
b
c
B is between A and C if and only if A, B and C
are distinct collinear points (on ABC) and
AB + BC = AC
AB = | b – a | = b – a
BC = | c – b | = c – b
AB + BC = (b – a) + (c – b) = c – a = AC
Mr. Chin-Sung Lin
ERHS Math Geometry
Line Segment
A
B
Segment
A segment is a subset, or a part of a line
consisting of two endpoints and all points on
the line between them
Symbol: AB
Mr. Chin-Sung Lin
ERHS Math Geometry
Line Segment
Length or Measure of a Line Segment
A
B
The length or measure of a line segment is the
distance between its endpoints, i.e., the
absolute value of the difference of the
coordinates of the two points
AB = |a - b| = |b - a|
Symbol: AB
Mr. Chin-Sung Lin
ERHS Math Geometry
Line Segment
Length or Measure of a Line Segment
A
B
AB represents segment AB
AB represents the measure of AB
Mr. Chin-Sung Lin
ERHS Math Geometry
Line Segment
Congruent Line Segments
A
C
B
D
Congruent segments are segments that have
the same measure
Mr. Chin-Sung Lin
ERHS Math Geometry
Line Segment
Congruent Line Segments
A
C
B
D
AB ≅ CD, the segments are congruent
AB = CD, the measures/distances are the same
Mr. Chin-Sung Lin
ERHS Math Geometry
Midpoints &
Bisectors
Mr. Chin-Sung Lin
ERHS Math Geometry
Line Segment
Midpoint of a Line Segment
A
M
B
The midpoint of a line segment is a point of that
line segment that divides the segment into two
congruent segments
Mr. Chin-Sung Lin
ERHS Math Geometry
Line Segment
Midpoint of a Line Segment
A
AM ≅ MB
AM = (1/2) AB
AB = 2AM
or
or
or
M
B
AM = MB
MB = (1/2) AB
AB = 2MB
Mr. Chin-Sung Lin
ERHS Math Geometry
Line Segment
Midpoint of a Line Segment
A
a
M
B
b
Coordinate of the midpoint of AB is (a + b)/2
Midpoint is the average point
Mr. Chin-Sung Lin
ERHS Math Geometry
Line Segment
Bisector of a Line Segment
D
E
M
A
B
C
F
The bisector of a line segment is any line or
subset of a line that intersects the segment at
its midpoint
Mr. Chin-Sung Lin
ERHS Math Geometry
Line Segment
Adding/Subtracting
Line Segments
A
a
P
B
b
A line segment, AB is the sum of two line
segments, AP and PB, if P is between A and B
AB = AP + PB
AP = AB – PB
PB = AB - AP
Mr. Chin-Sung Lin
ERHS Math Geometry
Rays & Angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Half-Lines and Rays
On one side of a point
P
A
B
Two points, A and B, are on one side of a point
P if A, B, and P are collinear and P is not
between A and B
Mr. Chin-Sung Lin
ERHS Math Geometry
Half-Lines and Rays
Half-Line
P
Half-line
A
B
Half-line
A half-line consists of the set of all points on
one side of a point of division, not including
that point (endpoint)
Mr. Chin-Sung Lin
ERHS Math Geometry
Half-Lines and Rays
Ray
A
B
A ray is the part of a line consisting of a point
on a line and all the points on one side of the
point (endpoint)
A ray consists of an endpoint and a half-line
Mr. Chin-Sung Lin
ERHS Math Geometry
Half-Lines and Rays
Ray
A
B
A ray AB is written as AB, where A needs to be
the endpoint
Mr. Chin-Sung Lin
ERHS Math Geometry
Half-Lines and Rays
Opposite Rays
A
The opposite rays are two collinear rays with a
common endpoint, and no other point in
common
Opposite rays always form a line
Mr. Chin-Sung Lin
ERHS Math Geometry
Lines
A
Parallel Lines
C
B
D
Lines that do not intersect may or may not be
coplanar
Parallel lines are coplanar lines that do not
intersect
Segments and rays are parallel if they lie in
parallel lines
Mr. Chin-Sung Lin
ERHS Math Geometry
Lines
Skew Lines
C
D
Skew lines do not lie in the same plane
They are neither parallel nor intersecting
Mr. Chin-Sung Lin
ERHS Math Geometry
Basic Definition of
Angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Basic Definition
 Definition of Angles
 Naming Angles
 Degree Measure of Angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Definition of Angles
An angle is the union of two rays having the
same endpoints
The endpoint is called the vertex of an angle;
the rays are called the sides of the angle
Vertex: A
C
Sides: AB and AC
1
A
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Naming Angles
 Three letter: CAB or BAC
 A number (or lowercase letter) in the
interior of angle: 1
 A single capital letter (its vertex): A
C
exterior of angle
1
A
exterior of angle
interior of angle
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Naming Angle
X
O
Y
Mr. Chin-Sung Lin
ERHS Math Geometry
Naming Angle - XOY or YOX
X
O
Y
Mr. Chin-Sung Lin
ERHS Math Geometry
Degree Measure of Angles
Let OA and OB be opposite rays in a plane.
OA, OB and all the rays with endpoints O
that can be drawn on one side of AB can be
paired with the real numbers from 0 to 180
in such a way that:
1. OA is paired with 0 and OB is paired with
180
A
O
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Degree Measure of Angles
2. If OC is paired with x and OD is paired with
y, then, the degree measure of the angle:
m COD = | x – y |
C
A
D
O
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Degree Measure of Angles
If OC is paired with 60 and OD is paired with
150, then, the degree measure of the angle:
m COD = ?
C
A
D
O
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Degree Measure of Angles
If OC is paired with 60 and OD is paired with
150, then, the degree measure of the angle:
m COD = | 60 – 150 | = | -90 | = 90.
C
A
D
O
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Type of Angles by Measures
 Straight Angle
 Obtuse Angle
 Right Angle
 Acute Angle
Mr. Chin-Sung Lin
ERHS Math Geometry
Straight Angle
A straight angle is an angle that is the union of
opposite rays
m AOB = 180
A
O
B
Mr. Chin-Sung Lin
ERHS Math Geometry
A Degree
A degree is the measure of an angle that is
1/180 of a straight angle
A
O
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Obtuse Angle
An obtuse angle is an angle whose degree
measure is greater than 90 and less than
180
90 < m DOE < 180
D
O
E
Mr. Chin-Sung Lin
ERHS Math Geometry
Right Angle
A right angle is an angle whose degree measure
is 90
m GHI = 90
G
H
I
Mr. Chin-Sung Lin
ERHS Math Geometry
Acute Angle
An acute angle is an angle whose degree
measure is greater than 0 and less than 90
0 < m DOE < 90
D
O
E
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Angles
Congruent angles are angles that have the same
measure
~ ABC
DOE =
m DOE = m ABC
D
O
A
E
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Bisector of an Angle
A bisector of an angle is a ray whose
endpoint is the vertex of the angle, and that
divides the angle into two congruent angles
If OC is the bisector of AOD
m AOC = m COD
A
C
O
D
Mr. Chin-Sung Lin
ERHS Math Geometry
Calculate Angle
C
A
B
O
If mAOB = 120, OC is an angle bisector,
then mAOC = ?
Mr. Chin-Sung Lin
ERHS Math Geometry
Calculate Angle
C
A
B
O
If mAOB = 120, OC is an angle bisector,
then mAOC = 60
Mr. Chin-Sung Lin
ERHS Math Geometry
Calculate Angle
C
B
A
O
If mCOB = 30, OC is an angle bisector,
then mAOB = ?
Mr. Chin-Sung Lin
ERHS Math Geometry
Calculate Angle
C
B
A
O
If mCOB = 30, OC is an angle bisector,
then mAOB = 60
Mr. Chin-Sung Lin
ERHS Math Geometry
Adding Angles
A non-straight angle AOC is the sum of two
angles AOP and POC if point P is in the
interior of angle AOC
A
P
AOC = AOP + POC
O
C
Note that AOC may be a straight angle with P
any point not on AOC
Mr. Chin-Sung Lin
ERHS Math Geometry
Calculate Angle
C
A
B
O
If mAOC = 50, mBOC = 40,
then mAOB = ?
Mr. Chin-Sung Lin
ERHS Math Geometry
Calculate Angle
C
A
B
O
If mAOC = 50, mBOC = 40,
then mAOB = 90
Mr. Chin-Sung Lin
ERHS Math Geometry
Solve for x
C
B
A
O
OC is an angle bisector. If mAOB = 60,
mCOB = 2x,
then x = ?
Mr. Chin-Sung Lin
ERHS Math Geometry
Solve for x
C
B
A
O
OC is an angle bisector. If mAOB = 60,
mCOB = 2x,
then x = 15
Mr. Chin-Sung Lin
ERHS Math Geometry
Perpendicular Lines
Perpendicular lines are two lines that intersect
to form right angles
A
O
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Distance from a Point to a Line
Distance from a point to a line is the length of
the perpendicular from the point to the line
Mr. Chin-Sung Lin
ERHS Math Geometry
Triangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Polygons
A polygon is a closed figure in a plane that is
the union of line segments such that the
segments intersect only at their endpoints
and no segments sharing a common
endpoint are collinear
Mr. Chin-Sung Lin
ERHS Math Geometry
Triangles
A triangle is a polygon that has exactly three
sides
∆ ABC
Vertex:
Angle:
Side:
Length of
A
b
c
A, B, C
A, B, C
AB, BC, CA
B
a
side: AB = c, BC = a, AC = b
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Type of Triangles by Sides
 Scalene Triangles
 Isosceles Triangles
 Equilateral Triangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Scalene Triangle
A scalene triangle is a triangle that has no
congruent sides
B
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Isosceles Triangle
A isosceles triangle is a triangle that has two
congruent sides
B
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Equilateral Triangle
A equilateral triangle is a triangle that has three
congruent sides
B
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Parts of an Isosceles Triangle
Leg: the two congruent sides
Base: the third non-congruent side
Vertex Angle: the angle formed by the two
congruent side
Base Angle: the angles whose vertices are the
endpoints of the base
B
Vertex Angle
Leg
Base Angle
Leg
A
Base
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Type of Triangles by Angles
 Acute Triangle
 Right Triangle
 Obtuse Triangle
 Equiangular Triangle
Mr. Chin-Sung Lin
ERHS Math Geometry
Acute Triangle
An acute triangle is a triangle that has three
acute angles
B
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Right Triangle
An right triangle is a triangle that has a right
angle
B
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Obtuse Triangle
An obtuse triangle is a triangle that has an
obtuse angle
B
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Equiangular Triangle
An equiangular triangle is a triangle that has
three congruent angles
B
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Parts of a Right Triangle
Leg: the two sides that form the right angle
Hypotenuse: the third side opposite the right
angle
B
Hypotenuse
A
Leg
Right Angle
C
Leg
Mr. Chin-Sung Lin
ERHS Math Geometry
Included Sides
If a line segment is the side of a triangle, the
endpoints of that segment is the vertics of two
angles, then the segment is included between
those two angles
AB is included between A and B
BC is included between B and C
CA is included between C and A
B
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Included Angles
Two sides of a triangle are subsets of the rays of an
angle, and the angle is included between those
sides
A is included between AB and AC
B is included between AB and BC
C is included between BC and AC
B
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Opposite Sides / Angles
For each side of a triangle, there is one vertex of the
triangle that is not the endpoint of that side
A is opposite to BC and BC is opposite to A
B is opposite to CA and CA is opposite to B
C is opposite to AB and AB is opposite to C
B
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Using Diagrams in Geometry
We may assume:
 A line segment is part of a line
 An intersect point is a point on both lines
 Points on a segment are between endpoints
 Points on a line are collinear
 A ray in the interior of an angle with its endpoint at
the vertex of the angle separate the angle into two
adjacent angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Using Diagrams in Geometry
We may NOT assume:

One segment is longer, shorter or equal to another one

A point is a midpoint of a segment

One angle is greater, smaller or equal to another one

Lines are perpendicular or angles are right angles

A triangle is isosceles or equilateral

A quadrilateral is a parallelogram, rectangle, square,
rhombus, or trapezoid
Mr. Chin-Sung Lin
ERHS Math Geometry
Q&A
Mr. Chin-Sung Lin
ERHS Math Geometry
The End
Mr. Chin-Sung Lin