MATHEMATICS 8

Remedial Activity
Quarter 2 ( Week 7 & 8)
The if-then statements, in terms of p and q, can be converted into inverse, converse and
contrapositive forms.
Conditional Statement If p, then q
Inverse If not p, then not q
Converse If q, then p
Contrapositive If not q, then not p
Inverse- To write the inverse of a conditional statement, simply negate both the hypothesis and
conclusion.
Conditional statement ( If p, then q ) Inverse (If not p, then q )
If you drink water, then you obey your thirst. If you do not drink water, then you do not
obey your thirst.
Converse - To write the converse of a conditional statement, simply interchange the hypothesis
and the conclusion. That is, the then part becomes the if part and the if part becomes the then
part.
Conditional statement ( If p, then q ) Converse ( if p, then q )
If you drink water, then you obey your thirst. If you obey your thirst, then you drink
water.

Contrapositive - To form the contrapositive of a conditional statement, first, get its inverse.
Then, interchange its hypothesis and conclusion.
Conditional statement ( If p, Inverse (If not p, then q ) Contrapositive ( if not q,
then q ) then not p )
If you drink water, then you If you do not drink water, If you do not obey your
obey your thirst. then you do not obey your thirst, then you do not drink
thirst. water.
Using Inductive or Deductive Reasoning in an Argument
Inductive reasoning- gathers specific information usually through observation and
measurement, formulate conjecture/s, then draw generalization or conclusion based on the
carefully gathered information.
Examples:
1. In the sequence, 10, 20, 30, …, the next term is 40.
2. John, a math challenger, is good in mathematics. Joan, Josh, and Bea who are also math
challengers are good in mathematics. Therefore, all math challengers are good in mathematics.
3. The chair in the living room is red. The chair in the dining room is red. Therefore, the color
of the chairs in the house is red.
Inductive reasoning allows you to make a general rule from specific examples.
Like in example 1, you are given a sequence with first three terms are 10, 20 and 30. From
these specific examples, you may then generalize that the sequence is a sequence of numbers
that are divisible by or multiple of 10. Hence, you conclude that the next term is 40.

Deductive reasoning- uses acceptable facts, proven theorem as proof to draw a specific case
or situation.
1. Sally does not drink soft drinks. Then, it follows that she does not drink Cola.
2. All numbers ending in 0 or 5 are divisible by 5. Number 35 ends with 5. Therefore, it must
be divisible by 5.
Deductive reasoning starts from a general statement or fact to conclude into specific example
or claim. For instance, in example 1, you are given that Sally does not drink soft drinks. Since
a cola is a specific example of a soft drink, then it follows that Sally does not drink cola.

Activity 1. Rewrite the given statement into converse, inverse, and contrapositive. (2
points each).
Statement: If two lines are parallel, then they never intersect.

Converse:___________________________________________________
Inverse:_____________________________________________________
Contrapositive:________________________________________________

Activity 2. Determine the converse, inverse, and contrapositive of the given statement. (1
point each).

Statement: If an angle measures 90o, then it is a right angle.

_________________1. If an angle is not right, then it does not measure 90o.
_________________2. If an angle is right, then it measures 90o.
_______________3. If an angle does not measure 90o, then it is not right angle.

Activity 3. Draw a conclusion from each given situation below (2 points each).

1. Filipinos are peace loving people, Veronica is a Filipino, therefore

________________________________________________________________________.
2. Coplanar points are points on the same plane. X, Y, Z are coplanar, therefore,
________________________________________________________________________.
3. 1, 4, 9, 25, 36. The next number is _______.

Direction: Read and understand the questions carefully. Choose the letter that
corresponds to the correct answer.
1. The converse of the statement: “If I eat a pint of ice cream, then I will gain weight” is.
A. If I will gain weight, then I will eat a pint of ice cream.
B. If I do not eat a pint of ice cream, then I will not gain weight.
C. If I will not gain weight, then I do not eat a pint of ice cream.
D. If I do not eat a pint of ice cream, then I will gain weight.

2. If m∠A + m∠B = 90, then

A. ∠A ≅ ∠B
B. ∠A and ∠B are right angles
C. ∠A and ∠B are complementary
D. ∠A and ∠B are supplementary

3. What is the Contrapositive of the statement “If the number is divisible by 2 and 3, then it is
divisible by 6.”
A. If the number is divisible by 6, then it is divisible by 2 and 3.
B. If the number is not divisible by 2 and 3, then it is not divisible by 6.
C. If the number is not divisible by 6, then it is not divisible by 2 and 3.
D. If the number is divisible by 2 and 3, then it is not divisible by 6.

4. Which of the following statements is true?

A. If ∠1 has a measure of 90, then ∠1 is obtuse.
B. If ∠1 has a measure of 140, then ∠1 is acute.
C. If ∠1 has a measure of 35, then ∠1 is acute.
D. If ∠1 has a measure of 180, then ∠1 is right.

5. Rewrite the statement into its Converse “If two segments are congruent, then they have the
same measure”.
A. If two segments have the same length, then they are congruent.
B. If two segments have not the same length, then they are not congruent.
C. If two segments are not congruent, then they have not the same measure.
D. If two segments have the same measure, then they are congruent.

6. Which of the following best described deductive reasoning?

A. Using logic to draw conclusion based on accepted statements.
B. Accepting the meaning of a term without definition.
C. Defining mathematical terms in relation to physical objects.
D. Inferring a general truth by examining a number of specific examples.

7. 15, 30, 45, 60, the next number is ________________.

A. 65
B. 75
C. 85
D. 90

8. What conclusion can you draw out of the given statements below. Filipinos are hospital,
Peter is a Filipino.
A. Therefore, Peter is hospitable.
B. Therefore, Filipinos are Peter.
C. Therefore, hospitable is a Filipino.
D. Peter is a Filipino.

For numbers 9-10. Determine whether inductive or deductive reasoning is applied in each case.
______________9. Right angle are congruent. Angle A and angle B are right angles then angle
A and angle B are congruent.

______________10. Mylene randomly picked ten green marbles and she concludes that all
marbles must be green.
 ENHANCED MATHEMATICS 8
Remedial Activity
Quarter 2 ( WeeK 8)
“Writing Proofs (Direct and Indirect)”

Properties of Equality
Let 𝑎, 𝑏, and 𝑐 be real numbers.

Properties of Inequality
Let 𝑎, 𝑏, and 𝑐 be real numbers.

Postulates

A proof is a logical argument in which each statement you make is

supported/justified by given information, definitions, axioms, postulates, theorems, and
previously proven statements.
Note:
• An axiom is any mathematical statement that serves as a starting point from which other points
are logically derived.
• A postulate is a statement that is accepted without proof.
• A theorem is a statement accepted after it is proved deductively.
There are two methods in writing proof. These include:
1. Direct Proof
2. Indirect Proof
Direct proof is a type of logical reasoning that uses accepted facts to reason in a step-by-step
manner until the desired statement is obtained.
Steps in writing a direct proof:
1. Assume the statement 𝑝 is true.
2. Use what you know about 𝑝 and other facts as necessary to deduce that another statement 𝑞 is
true, that is to show 𝑝→𝑞 (if p then q) is true.
Two-Column Form is one way of organizing a proof. This consists of two columns, one for
statements and one for reasons.

Illustrative Example 1.
If 2 (3𝑥 + 4) = 56, then 𝑥 = 8.
Given: 2 (3𝑥 + 4) = 56
Prove: 𝑥 = 8

Indirect proof is a type of proof in which a statement to be proved is assumed false (by
negation) and if the assumption leads to an impossibility, then the statement assumed false has
been proved to be true.
Steps in writing an indirect proof:
1. Identify the statement you want to prove. Assume temporarily that this statement is false by
assuming that its opposite is true.
2. Reasons logically until you reach a contradiction.
3. Point out that the desired conclusion must be true because the contradiction proves the
temporary assumption false.

Indirect proof is a type of proof in which a statement to be proved is assumed false (by negation) and if the
assumption leads to an impossibility, then the statement assumed false has been proved to be true.
Steps in writing an indirect proof:
1. Identify the statement you want to prove. Assume temporarily that this statement is false by assuming that its
opposite is true.
2. Reasons logically until you reach a contradiction.
3. Point out that the desired conclusion must be true because the contradiction proves the temporary assumption
false.

But 𝑥 = 4 contradicts the given statement that 𝑥 = 2. Since our assumption is false therefore, 3𝑥 − 4 ≠ 8 is true.
Activity 1: I Need Proof!
Directions: Fill in the blanks by choosing your answers from the box. Write it on a separate sheet of paper.

Prove that if 2(𝑏 + 1) = −6, then 𝑏 = −4.

Given:2(𝑏 + 1) = −6
Prove: 𝑏 = −4

Activity 2: Prove Me Indirectly!

Directions: Arrange the statements to its logical order using an indirect proof. Use letters A to F. Write your
answers on a separate sheet of paper.
Prove that if 𝑦 = −3, then 4(𝑦 − 3) ≠ −20.
_________1. 4𝑦 − 12 = −20 by distributive property of equality.
_________2. Hence our assumption is false and 4(𝑦 − 3) ≠ −20 is true
_________3. Using addition property of equality, 4𝑦 − 12 + 12 = −20 +12.So 4𝑦 = −8 by simplifying.
_________4. Assume that 4(𝑦 − 3) = −20. Take this statement as true and solve for 𝑦.
_________5. Thus, division property of equality, So 𝑦 = −2 by simplifying.
_________6. But 𝑦 = −2, contradicts the given statement that 𝑦 = −3.
Directions: Choose the letter of the correct answer. Write the chosen letter on a
separate sheet of paper.
1. What is an indirect proof?
A. A proof that always involves the multiplication of two values.
B. A proof that can only use number properties to show that a certain statement is false.
C. A proof that assumes a statement's hypothesis is true and uses a series of logical deductions to conclude that
the statement's conclusion is true
D. A proof that assumes that the statement being proven is false and then attempts to find a contradiction to that
assumption proving the original statement to be true
2. A proof that 𝑝 → 𝑞 is true based on the fact that 𝑞 is true, such proof is known as _____.
A. Direct Proof
B. Indirect Proof
C. Both A and B are correct
D. Neither A nor B are correct
3. Which of the following occurs with a direct proof?
A. Statements are supported by known facts and definitions.
B. A conditional statement is proven.
C. A series of statements are made.
D. All statements are correct.
4. Which statement is NOT true?
A. Start an indirect proof by assuming that the statement is true.
B. Start an indirect proof by assuming that the statement is false.
C. Use direct proof through accepted facts to reason in a step-by-step
manner until the desired statement is arrived.
D. Use two-column form to show a way of organizing a proof. This form
consists of two columns, one for statements and one for reasons.
5. What assumption would you make from the statement that ̅𝐴𝐵 ̅ ̅ is congruent to ̅𝐵𝐶
̅ ̅ using indirect proof?
A. 𝐴 ≠ 𝐵
B. 𝐵 ≠ 𝐶
C. 𝐴𝐵 ≠ 𝐵𝐶
̅ ̅ ≤ ̅𝐶𝐵
D. ̅𝐵𝐴 ̅
Consider the statement for your answers in numbers 6 and 7.
Given: 𝐴, 𝐵, 𝐶, and 𝐷 are collinear in that order and 𝐴𝐵 = 𝐶𝐷. Prove: 𝐴𝐶 = 𝐵𝐷

6. What is the “reason” for the "statement" 𝐵𝐶 = 𝐵𝐶 in Step 3 of the proof?

A. Commutative Property
B. Reflexive Property
C. Symmetric Property
D. Transitive Property

7. What is the “reason” from the "statement" 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶, 𝐵𝐶 + 𝐶𝐷 = 𝐵𝐷 in Step 5 of the proof?
A. Angle Addition Postulate
B. Segment Addition Postulate
C. Commutative Property of Equality
D. Distributive Property of Equality

8. Which statement is true using Subtraction Property of Equality in Step 3?

A. 3𝑦 + 4 − 10 = 10 − 10
B. 3𝑦 + 4 − 4 = 10 − 4
C. 3𝑦 + 4 + 10 = 10 + 10
D. 3𝑦 + 4 + 4 = 10 + 4

9. What is the reason for the statement 𝑦 = 2 in Step 4 of the proof?

A. Addition Property of Equality
B. Multiplication Property of Equality
C. Subtraction Property of Equality
D. Substitution Property

10. What assumption would you make from the statement that ⊿𝐴𝐵𝐶 ≅ ⊿𝐷𝐸𝐹 using
indirect proof?
A. ⊿𝐴𝐵𝐶 ≠ ⊿𝐷𝐸𝐹
B. ⊿𝐴𝐵𝐶 ≇ ⊿𝐷𝐸𝐹
C. ⊿𝐴𝐵𝐶 ≤ ⊿𝐷𝐸𝐹
D. ⊿𝐴𝐵𝐶 ≥ ⊿𝐷𝐸𝐹