Essentials of Geometry

Eleanor Roosevelt High School

Geometry

Mr. Chin-Sung Lin

Basic Definitions

ERHS Math Geometry

Mr. Chin-Sung Lin

Definition

ERHS Math Geometry

Mr. Chin-Sung Lin

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

Postulate

ERHS Math Geometry

Mr. Chin-Sung Lin

A postulate is an accepted statement of fact

Undefined Terms:Set, Point, Line & Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

Undefined Terms

Set

ERHS Math Geometry

Mr. Chin-Sung Lin

A collection of objects such that it is possible to determine whether a given object belongs to the collection or not

Undefined Terms

Point

ERHS Math Geometry

Mr. Chin-Sung Lin

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

A C D EB

Line

ERHS Math Geometry

Mr. Chin-Sung Lin

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

A B

Undefined Terms

Line

ERHS Math Geometry

Mr. Chin-Sung Lin

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

A B

Undefined Terms

Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

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”

R

Undefined Terms

Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

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

A

BC

Undefined Terms

Postulate

ERHS Math Geometry

Mr. Chin-Sung Lin

Through any two points there is exactly one line

A B

Undefined Terms

Postulate

ERHS Math Geometry

Mr. Chin-Sung Lin

If two lines intersect, then they intersect in exactly one point

P

Undefined Terms

Postulate

ERHS Math Geometry

Mr. Chin-Sung Lin

If two planes intersect, then they intersect in exactly a line

Undefined Terms

Properties of Real Numbers

ERHS Math Geometry

Mr. Chin-Sung Lin

Addition & Multiplication Operation Properties

ERHS Math Geometry

Mr. Chin-Sung Lin

Closure

Commutative Property

Associative Property

Identity Property

Inverse Property

Distributive Property

Multiplication Property of Zero

Closure

ERHS Math Geometry

Mr. Chin-Sung Lin

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

Commutative Property

ERHS Math Geometry

Mr. Chin-Sung Lin

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

Associative Property

ERHS Math Geometry

Mr. Chin-Sung Lin

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)

Identity Property

ERHS Math Geometry

Mr. Chin-Sung Lin

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

Inverse Property

ERHS Math Geometry

Mr. Chin-Sung Lin

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)

Distributive Property

ERHS Math Geometry

Mr. Chin-Sung Lin

Multiplication distributes over addition

a (b + c) = a b + a c

(a + b) c = a c + b c

Multiplication Property of Zero

ERHS Math Geometry

Mr. Chin-Sung Lin

Zero has no multiplicative inverse

Zero product property

a b = 0 if and only if a = 0 or b = 0

Exercise

ERHS Math Geometry

Mr. Chin-Sung Lin

Identify the additive and multiplicative inverses of the following nonzero real numbers:

9

-6

d

-b

(3 – b)

Exercise

ERHS Math Geometry

Mr. Chin-Sung Lin

Identify the additive and multiplicative inverses of the following nonzero real numbers:

9 -9

-6 6

d -d

-b b

(3 – b) (b – 3)

Exercise

ERHS Math Geometry

Mr. Chin-Sung Lin

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)

Exercise

ERHS Math Geometry

Mr. Chin-Sung Lin

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

Exercise

ERHS Math Geometry

Mr. Chin-Sung Lin

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)

Lines & Line Segments

ERHS Math Geometry

Mr. Chin-Sung Lin

Distance between Tow Points

Distance

ERHS Math Geometry

Mr. Chin-Sung Lin

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 |

A B

a b

Order of Points

Betweenness

ERHS Math Geometry

Mr. Chin-Sung Lin

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 – aBC = | c – b | = c – bAB + BC = (b – a) + (c – b) = c – a = AC

A B

a b

C

c

Line Segment

Segment

ERHS Math Geometry

Mr. Chin-Sung Lin

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

A B

Line SegmentLength or Measure of a Line Segment

ERHS Math Geometry

Mr. Chin-Sung Lin

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

A B

ERHS Math Geometry

Mr. Chin-Sung Lin

AB represents segment ABAB represents the measure of AB

Line SegmentLength or Measure of a Line Segment

A B

Line Segment

Congruent Line Segments

ERHS Math Geometry

Mr. Chin-Sung Lin

Congruent segments are segments that have the same measure

A B

C D

Line Segment

Congruent Line Segments

ERHS Math Geometry

Mr. Chin-Sung Lin

A B

C D

AB CD, the segments are congruentAB = CD, the measures/distances are the same

≅

Midpoints & Bisectors

ERHS Math Geometry

Mr. Chin-Sung Lin

Line Segment

Midpoint of a Line Segment

ERHS Math Geometry

Mr. Chin-Sung Lin

The midpoint of a line segment is a point of that line segment that divides the segment into two congruent segments

A BM

Line Segment

Midpoint of a Line Segment

ERHS Math Geometry

Mr. Chin-Sung Lin

AM MB or AM = MBAM = (1/2) AB or MB = (1/2) AB

AB = 2AM or AB = 2MB

A BM

≅

Line Segment

Midpoint of a Line Segment

ERHS Math Geometry

Mr. Chin-Sung Lin

Coordinate of the midpoint of AB is (a + b)/2

Midpoint is the average point

A BM

a b

Line SegmentBisector of a Line Segment

ERHS Math Geometry

Mr. Chin-Sung Lin

The bisector of a line segment is any line or subset of a line that intersects the segment at its midpoint

A BM

C

DE

F

Line Segment

Adding/Subtracting Line Segments

ERHS Math Geometry

Mr. Chin-Sung Lin

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

A BP

a b

Rays & Angles

ERHS Math Geometry

Mr. Chin-Sung Lin

Half-Lines and Rays

On one side of a point

ERHS Math Geometry

Mr. Chin-Sung Lin

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

P BA

Half-Lines and Rays

Half-Line

ERHS Math Geometry

Mr. Chin-Sung Lin

A half-line consists of the set of all points on one side of a point of division, not including that point (endpoint)

P BA

Half-lineHalf-line

Half-Lines and Rays

Ray

ERHS Math Geometry

Mr. Chin-Sung Lin

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

A B

Half-Lines and Rays

Ray

ERHS Math Geometry

Mr. Chin-Sung Lin

A ray AB is written as AB, where A needs to be the endpoint

A B

Half-Lines and Rays

Opposite Rays

ERHS Math Geometry

Mr. Chin-Sung Lin

The opposite rays are two collinear rays with a common endpoint, and no other point in common

Opposite rays always form a line

A

Lines

Parallel Lines

ERHS Math Geometry

Mr. Chin-Sung Lin

Lines that do not intersect may or may not be coplanar Parallel lines are coplanar lines that do not intersectSegments and rays are parallel if they lie in parallel lines

A B

C D

Lines

Skew Lines

ERHS Math Geometry

Mr. Chin-Sung Lin

Skew lines do not lie in the same plane

They are neither parallel nor intersecting

AB

C D

Basic Definition of Angles

ERHS Math Geometry

Mr. Chin-Sung Lin

Basic Definition

ERHS Math Geometry

Mr. Chin-Sung Lin

Definition of Angles Naming Angles Degree Measure of Angles

Definition of Angles

ERHS Math Geometry

Mr. Chin-Sung Lin

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

Sides: AB and AC

BA

C

1

Naming Angles

ERHS Math Geometry

Mr. Chin-Sung Lin

Three letter: CAB or BAC A number (or lowercase letter) in the interior of angle: 1 A single capital letter (its vertex): A

AB

C

interior of angle

exterior of angle

exterior of angle

1

Naming Angle

ERHS Math Geometry

Mr. Chin-Sung Lin

OY

X

Naming Angle - XOY or YOX

ERHS Math Geometry

Mr. Chin-Sung Lin

OY

X

Degree Measure of Angles

ERHS Math Geometry

Mr. Chin-Sung Lin

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 BO

Degree Measure of Angles

ERHS Math Geometry

Mr. Chin-Sung Lin

2. If OC is paired with x and OD is paired with y, then, the degree measure of the angle: m COD = | x – y |

A BO

DC

Degree Measure of Angles

ERHS Math Geometry

Mr. Chin-Sung Lin

If OC is paired with 60 and OD is paired with 150, then, the degree measure of the angle: m COD = ?

A BO

DC

Degree Measure of Angles

ERHS Math Geometry

Mr. Chin-Sung Lin

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.

A BO

DC

Type of Angles by Measures

ERHS Math Geometry

Mr. Chin-Sung Lin

Straight Angle

Obtuse Angle

Right Angle

Acute Angle

Straight Angle

ERHS Math Geometry

Mr. Chin-Sung Lin

A straight angle is an angle that is the union of opposite rays m AOB = 180

A BO

A Degree

ERHS Math Geometry

Mr. Chin-Sung Lin

A degree is the measure of an angle that is 1/180 of a straight angle

A BO

Obtuse Angle

ERHS Math Geometry

Mr. Chin-Sung Lin

An obtuse angle is an angle whose degree measure is greater than 90 and less than 180 90 < m DOE < 180

EO

D

Right Angle

ERHS Math Geometry

Mr. Chin-Sung Lin

A right angle is an angle whose degree measure is 90 m GHI = 90

IH

G

Acute Angle

ERHS Math Geometry

Mr. Chin-Sung Lin

An acute angle is an angle whose degree measure is greater than 0 and less than 90 0 < m DOE < 90

EO

D

Congruent Angles

ERHS Math Geometry

Mr. Chin-Sung Lin

Congruent angles are angles that have the same measureDOE = ABC m DOE = m ABC

~

CB

A

EO

D

Bisector of an Angle

ERHS Math Geometry

Mr. Chin-Sung Lin

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 AODm AOC = m COD

DO

AC

Calculate Angle

ERHS Math Geometry

Mr. Chin-Sung Lin

If mAOB = 120, OC is an angle bisector, then mAOC = ?

B A

O

C

Calculate Angle

ERHS Math Geometry

Mr. Chin-Sung Lin

If mAOB = 120, OC is an angle bisector, then mAOC = 60

B A

O

C

Calculate Angle

ERHS Math Geometry

Mr. Chin-Sung Lin

If mCOB = 30, OC is an angle bisector, then mAOB = ?

BA

O

C

Calculate Angle

ERHS Math Geometry

Mr. Chin-Sung Lin

If mCOB = 30, OC is an angle bisector, then mAOB = 60

BA

O

C

Adding Angles

ERHS Math Geometry

Mr. Chin-Sung Lin

A non-straight angle AOC is the sum of two angles AOP and POC if point P is in the interior of angle AOC

AOC = AOP + POC

Note that AOC may be a straight angle with P any point not on AOC

CO

AP

Calculate Angle

ERHS Math Geometry

Mr. Chin-Sung Lin

If mAOC = 50, mBOC = 40, then mAOB = ?

B A

O

C

Calculate Angle

ERHS Math Geometry

Mr. Chin-Sung Lin

If mAOC = 50, mBOC = 40, then mAOB = 90

B A

O

C

Solve for x

ERHS Math Geometry

Mr. Chin-Sung Lin

OC is an angle bisector. If mAOB = 60, mCOB = 2x,

then x = ?

BA

O

C

Solve for x

ERHS Math Geometry

Mr. Chin-Sung Lin

OC is an angle bisector. If mAOB = 60, mCOB = 2x,then x = 15

BA

O

C

Perpendicular Lines

ERHS Math Geometry

Mr. Chin-Sung Lin

Perpendicular lines are two lines that intersect to form right angles

CO

A

Distance from a Point to a Line

ERHS Math Geometry

Mr. Chin-Sung Lin

Distance from a point to a line is the length of the perpendicular from the point to the line

C

O

A

Triangles

ERHS Math Geometry

Mr. Chin-Sung Lin

Polygons

ERHS Math Geometry

Mr. Chin-Sung Lin

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

Triangles

ERHS Math Geometry

Mr. Chin-Sung Lin

A triangle is a polygon that has exactly three sides

∆ ABCVertex: A, B, CAngle: A, B, CSide: AB, BC, CALength of side: AB = c, BC = a, AC = b

a

A

CB

bc

Type of Triangles by Sides

ERHS Math Geometry

Mr. Chin-Sung Lin

Scalene Triangles

Isosceles Triangles

Equilateral Triangles

Scalene Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

A scalene triangle is a triangle that has no congruent sides

A

C

B

Isosceles Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

A isosceles triangle is a triangle that has two congruent sides

A C

B

Equilateral Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

A equilateral triangle is a triangle that has three congruent sides

A C

B

Parts of an Isosceles Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

Leg: the two congruent sidesBase: the third non-congruent sideVertex Angle: the angle formed by the two

congruent sideBase Angle: the angles whose vertices are the

endpoints of the base

A C

B

Base

LegLegBase Angle

Vertex Angle

Type of Triangles by Angles

ERHS Math Geometry

Mr. Chin-Sung Lin

Acute Triangle

Right Triangle

Obtuse Triangle

Equiangular Triangle

Acute Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

An acute triangle is a triangle that has three acute angles

A C

B

Right Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

An right triangle is a triangle that has a right angle

A C

B

Obtuse Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

An obtuse triangle is a triangle that has an obtuse angle

A

C

B

Equiangular Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

An equiangular triangle is a triangle that has three congruent angles

A C

B

Parts of a Right Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

Leg: the two sides that form the right angleHypotenuse: the third side opposite the right

angle

Leg

Leg

Right Angle

A C

B

Hypotenuse

Included Sides

ERHS Math Geometry

Mr. Chin-Sung Lin

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

A C

B

Included Angles

ERHS Math Geometry

Mr. Chin-Sung Lin

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 ACB is included between AB and BC C is included between BC and AC

A C

B

Opposite Sides / Angles

ERHS Math Geometry

Mr. Chin-Sung Lin

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

A C

B

Using Diagrams in Geometry

ERHS Math Geometry

Mr. Chin-Sung Lin

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

Using Diagrams in Geometry

ERHS Math Geometry

Mr. Chin-Sung Lin

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

Q & A

ERHS Math Geometry

Mr. Chin-Sung Lin

The End

ERHS Math Geometry

Mr. Chin-Sung Lin