The document discusses logical equivalences between statements and their converses, inverses, and contrapositives. It provides examples of converting statements to if-then form and deriving their converses, inverses, and contrapositives by switching or negating the hypothesis and conclusion. The key steps are: the converse switches the hypothesis and conclusion; the inverse negates both; the contrapositive switches and negates. Understanding these logical equivalences is important for demonstrating understanding of formal arguments and proofs in mathematics.

1. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
G R A D E 8
M A T H E M A T I C S
EQUIVALENCE OF STATEMENT AND ITS
CONTRAPOSITIVE, COVERSE AND INVERSE

2. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
Content Standard:
The learner demonstrates understanding of axiomatic development of
geometry, triangle congruence, inequalities in triangles and parallel and
perpendicular lines.
Performance Standard:
The learner is able to engage in formal arguments, reasons and proof.

3. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
Review:
Convert each statement to if – then form, then identify the hypothesis
and the conclusion.
1. Good citizens obey rules and regulations.
2. Filipinos are God – Fearing People

4. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
1. Good citizens obey rules and regulations.
If we are good citizens, then we obey rules and regulations.
2. Filipinos are God – Fearing People
If we are Filipinos, then we are God – Fearing People.
Note: Words in red color are the hypothesis and words in blue are the
conclusion.

5. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
Motivation:
Direction: Tell something about triangles.
Possible Answer:
A triangle is a polygon.
Consider the answer above. We can convert it to if – then form, then we can
form, then we can form its converse, inverse and contrapositive.
Study the following table:

6. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
Statement If – then form Converse Inverse Contrapositive
A triangle is a
polygon.
If a shape is a
triangle, then it
is a polygon.
If a shape is a
polygon, then it
is a triangle.
If a shape is
not a triangle,
then it is not a
polygon.
If a shape is
not a polygon,
then it is not a
triangle.
Discuss with your group how the converse, inverse and contrapositive a given
statement is written.

7. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
Statement If – then form Converse Inverse Contrapositive
A triangle is a
polygon.
If a shape is a
triangle, then it
is a polygon.
If a shape is a
polygon, then it
is a triangle.
If a shape is
not a triangle,
then it is not a
polygon.
If a shape is
not a polygon,
then it is not a
triangle.
If hypothesis (p): If a shape is a triangle
Then conclusion (q) is : then it is a polygon.
What happen to p and q in the converse?

8. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
Statement If – then form Converse Inverse Contrapositive
A triangle is a
polygon.
If a shape is a
triangle, then it
is a polygon.
If a shape is a
polygon, then it
is a triangle.
If a shape is
not a triangle,
then it is not a
polygon.
If a shape is
not a polygon,
then it is not a
triangle.
If hypothesis (p): If a shape is a triangle
Then conclusion (q) is : then it is a polygon.
Compare the inverse and the original statement. What did you do with p?
What did you do with q?

9. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
Statement If – then form Converse Inverse Contrapositive
A triangle is a
polygon.
If a shape is a
triangle, then it
is a polygon.
If a shape is a
polygon, then it
is a triangle.
If a shape is
not a triangle,
then it is not a
polygon.
If a shape is
not a polygon,
then it is not a
triangle.
If hypothesis (p): If a shape is a triangle
Then conclusion (q) is : then it is a polygon.
Observe the changes in the contrapositive. Summarize your observation in
terms of p and q.

10. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
WHAT IS A CONVERSE OF A STATEMENT?

11. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
Switching the hypothesis and conclusion of a conditional statement
(if – then form).
Converse
Hypothesis (p): If a shape is a triangle
Conclusion (q) is : then it is a polygon.
Hypothesis (p): If a shape is a Polygon
Conclusion (q) is : then it is a triangle.
Converse
A triangle is a polygon.
Conditional Statement
Statement:

12. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
"If it is raining then the
grass is wet"
Example:
Note: As in the example, a proposition may be true but have a false converse.
Conditional Statement Converse
"If the grass is wet then it
is raining."

13. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
WHAT IS AN INVERSE OF A STATEMENT?

14. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
• Negating both the hypothesis and conclusion of a conditional statement
(if – then form).
Inverse
Hypothesis (p): If a shape is a triangle
Conclusion (q) is : then it is a polygon.
Hypothesis (p): If a shape is NOT a triangle
Conclusion (q) is : then it is NOT a polygon.
Inverse
Conditional Statement

15. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
"If it is raining, then the
grass is wet"
Example:
Note: As in the example, a proposition may be true but its inverse may be false.
Conditional Statement Inverse
" If it is not raining, then
the grass is not wet."

16. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
WHAT IS CONTRAPOSITIVE OF A
STATEMENT?

17. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
Switching the hypothesis and conclusion of a conditional and negating both.
Contrapositive
Hypothesis (p): If a shape is a triangle
Conclusion (q) is : then it is a polygon.
Hypothesis (p): If a shape is NOT a Polygon
Conclusion (q) is : then it is NOT a triangle.
Contrapositive
Conditional Statement

18. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
"If it is raining, then the
grass is wet"
Example:
Note: As in the example, the contrapositive of any true proposition is also true.
Conditional Statement Contrapositive
"If the grass is not wet,
then it is not raining."

19. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
We can summarize how to convert the statement in
terms of p and q. study the table below.
CONDITIONAL
STATEMENT
If p, then q.
CONVERSE If q, then p.
INVERSE If not p, then not q.
CONTRAPOSITIVE If not q, then not p.

20. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
STATEMENT An even number is divisible by two
CONDITIONAL
STATEMENT
CONVERSE
INVERSE
CONTRAPOSITIVE
A. Fill up the table below.

21. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
STATEMENT An even number is divisible by two
CONDITIONAL
STATEMENT
If a number is even, then it is divisible by two.
CONVERSE If a number is divisible by two, then it is even.
INVERSE If a number is not even, then it is not divisible by two.
CONTRAPOSITIVE If a number is not divisible by two, then it is not even.
A. Fill up the table below.

22. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
CONDITIONAL
STATEMENT
If two angles are congruent, then they have the same measure.
CONVERSE
INVERSE
CONTRAPOSITIVE
B. Fill up the table below.

23. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
CONDITIONAL
STATEMENT
If two angles are congruent, then they have the same measure.
CONVERSE If two angles have the same measure, then they are congruent.
INVERSE If two angles are not congruent, then they are not the same
measure.
CONTRAPOSITIVE If two angles are not the same measure, then they are not
congruent.
B. Fill up the table below.

24. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
Analyse the converse of the following statement. If it is
not true then give a counter example.
Conditional Statement: If a shape is a triangle, then it is a
polygon.
CONVERSE: If a shape is a polygon, then it is a triangle.
The converse is false, because a square is also a polygon, it is not necessarily
a triangle.

25. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
Analyse the converse of the following statement. If it is
not true then give a counter example.
Conditional Statement: If a number is even, then it is
divisible by two.
CONVERSE: If a number is divisible by two, then it is even.
The converse is true.

26. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
State the if – then form, converse, inverse and contrapositive of he following
statement.
1. Three non – collinear points determine a plane.
2. A rectangle has four right angles.
3. Perpendicular lines intersect.

27. Click to edit Master subtitle style
L.C. M8GE-IIg-2: Illustrates the equivalences of the statement and its G r a d e 8
Reference:
http://www.mathwords.com/c/converse.htm
http://www.mathwords.com/c/contrapositive.htm
http://www.mathwords.com/i/inverse_conditional.htm
https://www.catalysts.cc/en/the-catalysts-way/summary-of-how-it-all-started/
https://www.videoblocks.com/video/exercise-word-scale-physical-fitness-lose-weight-active-lifestyle-regimen-
h0cgkkhein9gkogy
http://www.differencebetween.info/difference-between-analyse-and-analyze
http://retiresuccessfully.co.za/test-yourself/