TOPIC 1: LINES AND ANGLES

TOPIC 1:LINES AND ANGLES

NOR SURIYA A.K SMG3013 SESSION 2021/202 1

Learning Outcomes

At the end of this topic, students should be able to:

1. Apply geometric concepts and construct geometric

proofs using both direct and indirect reasoning to form or

verify conjectures about plane figures.

2. Communicate mathematical ideas both in written and in

oral form.

NOR SURIYA A.K 2SMG3013 SESSION 2020/2021

1.1 SETS AND REASONING

Set is any collection of objects where the objects in

set are called as elements.

Let X and Y be two sets, then X is a subset of Y if

every element in X is also an element Y denoted

by . We also say that X is contained in Y

or Y contains X.

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1.1.1 SETS

X Y

SMG3013 SESSION 2020/2021

The union of X and Y, denoted by , is the set

of all elements which belong to X or to Y, that is

.

The intersection of X and Y, denoted by , is

the set of all elements which belong to both X

and Y, that is . If X and

Y do not have any elements in common, then X

and Y are said to be disjoint or nonintersecting,

denoted by .

Venn-diagram is a pictorial representation of sets

in which sets are represented by enclosed areas

on the plane. NOR SURIYA A.K 4

X Y

= { : or }X Y x x X x Y

X Y

= { : and }X Y x x X x Y

=X Y


Example 1:

1. X = {1,2,3} can be said as ‘X is a set of elements

1,2 and 3’ or ‘1,2 and 3 are the elements of set

X, denoted by ‘. The Venn-diagram for

X,

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1,2,3 X


2. A = {a,b,c,d}. So, and read as ‘e is not an

element in A’.

3. A = {1,2,3} and B = {counting number}. Then,

.

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e A

A B


4. T = {all triangles} and P = {all polygons}. Then,

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T P


1.1.2 STATEMENT

Statement is a set of words and/or symbols that

collectively make a claim that can be classified

as true or false.

Example 2:

1. 4 + 3 = 7 is a statement.

2. 7 < 3 is a statement.

3. Look out! is not a statement.

4. Where do you live? is not a statement


Negation of a statement is the opposite of the

original statement. If the given statement is true,

then its negation is false and vice versa.

Negation is denoted by ‘~’ and read as ‘not’. If P is

a statement then the negation of P is denoted by

~P.

The truth value of negation is as follows,


Example 3:

1. Given that P : 4 + 3 = 7. So, ~P : 4 + 3 ≠ 7.

2. Given that Q : All fish can swim. So, ~Q : Some

fish cannot swim.

Compound statement is a statement that

contained two or more statements separated by

logical connectors.


A conjunction is a compound statement formed by

using the word ‘and’, denoted by , to join two

simple statements. Let p and q be two

statements. The conjunction statement of p and

q is denoted by p q .

The truth value of p q is as follow,

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Example 4:

1. Let p : Amy played netball.

q : Ali played soccer.

p q : Amy played netball and Ali played

soccer.

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A disjunction is a compound statement formed by

using the word ‘or’, denoted by , to join two

simple statements. Let p and q be two

statements. The disjunction statement of p and q

is denoted by p q .

The truth value of p q is as follow,

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Example 5:

1. Let p : She runs slow.

q : He walks fast.

p q : She runs slow or he walks fast.

A conditional statement (or implication) is a

statement that is usually expressed with ‘If

…then …’. The conditional ‘if p then q’ is

denoted by and as well read as ‘p implies

q’.

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p q→


The truth value of is as follow,

Example 6:

1. Let p : Sarah is absent.

q : Sarah has a make-up assignment to

complete.

: If Sarah is absent then she has a make-

up assignment to complete.

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p q→

p q→


1.1.3 REASONING

Reasoning is a process based on experience and

principles that allows one to arrive at a

conclusion.

Three types of reasoning: a) intuition

b) induction

c) deduction


Intuition : an inspiration leading to the statement of

a theory.

Induction : an organized effort to test and validate

that theory.

Deduction : a formal argument that proves the

tested theory.


Law of Detachment

Let P and Q be simple statements and assume 1

and 2 are true. Then, a valid argument having

conclusion C has form

Otherwise, invalid argument.

Note that means ‘therefore’.

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Example 7:

Determine the following argument is valid or invalid.

1.

Solution:

Let p : It is raining.

q : Tim will stay in the house.

We have,

Thus, valid argument.


2.

Solution:

Let p : A man lives in Kuala Lumpur.

q : A man lives in Malaysia.

We have,

Thus, invalid argument.


1.2 GEOMETRY AND MEASUREMENT

Namely A, B and C are three points. When A, B

and C are on the same line, then they are said to

be collinear.

Line segment is part of a line that consists of two

distinct points on the line and all points between

them. Suppose line AB has a line segment, that

is indicated by .

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AB


From the diagram below, let be the line

segment of ABC . Then,

1. If AB = BC, then

a. Point B is the midpoint of AC.

b. and read as ‘ congruent to ’.

2. AC and DE are intersect at point B.

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AC

AB BC AB BC


Consider the diagram below. Then,

1. .

2. has been bisected, i.e. a line segment (or an

angle) is separated to two parts of equal measure

(Note : trisect means a line segment (or an angle) is

separated into three parts of equal measure).

3. is a straight angle, i.e. any angle having 1800

measure.

4. is a right angle, i.e. any angle having 900

measure.

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ABC CBD

DBF

EBF

ABD


Let figure below represent a circle.

1. Point O is the center of a circle.

2. OA is the radius (plural : radii) of a circle.

3. AB is an arc of a circle.


Exercise 1.2

Answer the following questions accordingly.

1. a) How many endpoints does a line segment have?

b) How many midpoints does a line segment have?

2.

Find

a) if and .

b) if and .

c) x if , and .

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ABC0

1 32 = 0

2 39 =

1068ABC =

1 2 =

1 x =2 2 3x = +

072ABC =


1.3 DEFINITION AND POSTULATES

Note that axiom or postulate is a statement that is

assumed to be true.

Theorem is a proved statement.

Four parts of mathematical system:


Characteristics of a good definition:


Postulate 1.3.1:

Through two distinct points, there is exactly one

line.

Postulate 1.3.2: Ruler Postulate

The measure of any line segment is a unique

positive number.

Postulate 1.3.3: Segment-addition Postulate

If X is a point on and A-X-B, then AX + XB =

AB.

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AB


Postulate 1.3.4:

If two lines intersect, they intersect at a point.

Postulate 1.3.5:

Through three noncollinear points, there is exactly

one plane.

Postulate 1.3.6:

If two distinct planes intersect, then their

intersection is a line.


Postulate 1.3.7:

Given two distinct points in a plane, the line

containing these points also lies in the plane.

Theorem 1.3.1:

The midpoint of a line is unique.

Postulate 1.3.8:

The measure of an angle is a unique positive

number.

Postulate 1.3.9: Angle-addition Postulate

If a point D lies in the interior of an angle ABC,

then .NOR SURIYA A.K 30

ABD DBC ABC + =SMG3013 SESSION 2020/2021

1.4 LINES AND ANGLES

Definition: Perpendicular lines

Perpendicular lines are two lines that meet to form

congruent adjacent angles. If two lines m and p

are perpendicular, they denoted by .

Theorem 1.4.1:

If two lines are perpendicular then they meet to

form right angles (right angle denoted by □).

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m p⊥


An angle is the union of two rays that share a

common endpoint.

Two angles are said to be adjacent if they have a

common vertex and a common side between

them.

Congruent angles are two angles with the same

measure.

The bisector of an angle is the ray that separates

the given angle into two congruent angles.


Two angles are complementary if the sum of their

measure is 900. Each angle in the pair is known

as the complement of other angle, i.e. .

Two angles are supplementary if the sum of the

measure is 1800. Each angle on the pair is

known as the supplement of other angle, i.e.

.

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0

1 2 90 + =

0

1 2 180 + =


Vertical angles are the pair of nonadjacent angles

in opposite directions when two straight lines are

intersect.

Theorem 1.4.2:

If two lines intersect, then the vertical angles

formed are congruent.


Types of angles:


Example 8:

Considering the diagram below,

a) and as well as and are vertical angles.

b) and are one of the adjacent angles.

c) and are congruent iff where A is the bisector.

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1

1

1

3 2

2

2

4

1 2 =


Example 9:

1. Given that . Find:

a. The complement of .

b. The supplement of .

Solution:

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0

1 30 =

1

1


2. Consider the diagram in Example 8. Find the value of ,

, and if,

Solution:

We know that . Then,

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1

2 3 4

0

1 2 180 + =


By Theorem 1.4.2, we know that vertical angles are

congruent, then

Using simultaneous equation of (1) and (2),


Substitute y = 200 into (1), then

Thus,


Exercise 1.4

1. Two angles are supplementary. One angle is 240 more

than twice the other. Find the measure of each angle.

2. Two angles are complementary. One angle is 120 larger

than the other. Find the size of each angle by solving a

system of equations.

3. Given that bisect where

i)

ii)

iii)

Find x and y.

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RSV

RST x y = +

2 2TSV x y = −064RSV =

ST


1.5 GEOMETRIC AND FORMAL

PROOFS

1.5.1 GEOMETRIC PROOF

Properties of equality


Further algebraic properties of equality


Example 10:

Given APB on . Prove that AP = AB – PB

Solution:

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AB


Properties of inequality


Example 11:

1. Given -4y > 30. Prove that .

Solution:

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15

2y

−


2. Given that . Prove that .

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1 22

3 3k −

1

2k


1.5.2 RELATIONS

Definition: Relations

Let A and B be sets. A binary relation, or simply relation

from A to B is a subset of , where the relation is

denoted by R.

For each pair of and , exactly one of the

following is true.

i) , we then say ‘a is R-related to b’, written as

aRb.

ii) , we then say ‘a is not R-related to b’, written

as .

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A B

a A b B

( , )a b R

( , )a b R

a Rb


In general, a relation connects two elements of an

associated set of objects.

Relation properties


An equivalence relation is a relation that has reflexive,

symmetric and transitive properties.

Congruence of angles (or line segments) is closely tied to

equality of angle measures (or line segment measures)

by the definition of congruence.

Properties for the congruence of angles


Example 12:

Consider the following diagram.

Given that intersects at O. Prove that .

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2 4 AC BD


Solution:


1.5.3 FORMAL PROOF

Essential parts of the formal proof of a theorem:

1. Statement : States the theorem to be proved.

2. Drawing : Represents the hypothesis of the theorem.

3. Given : Describes the Drawing according to the

information found in the hypothesis of the

theorem.

4. Prove : Describes the Drawing according to the

claim made in the conclusion of a theorem.

5. Proof : Order a list of claims (Statements) and

justification (Reasons), beginning with the Given

and ending with the Prove; there must be a

logical flow in this Proof.


Example 13:

Conduct a formal proof to prove that if two lines are

perpendicular, then they meet to form right angles.

Solution:



Let a statement be represented by ‘if p then q’. Then,

The converse of a statement is ‘if q then p’.

The inverse of a statement is ‘if not p then not q’.

The contrapositive of a statement is ‘if not q then not p’.

Example 14:

Statement : A square is a rectangle.

Converse : A rectangle is a square.

Inverse : If it is not a square then it is not a rectangle.

Contrapositive : If it is not a rectangle then it is not a square.


Exercise 1.5

1. Given that DEF on DF. Prove that DE = DF - EF.

2. Given that bisects . Prove that

3. Prove that if a = b and c = d, then a – c = b – d .



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ABCBD1

2 = ABD ABC

3 1(1 )

2 4x x− −

5

2x

22 ( 3)

5 3

xx− +

60

7 −x


6.

NOR SURIYA A.K 58SMG3013 SEM SESSION

2019/2020

TOPIC 1: LINES AND ANGLES

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