Conditional statements in geometry :
In this section, we are going to study a type of logical statement called conditional statement. A conditional statement has two parts, a hypothesis and a conclusion. If the statement is written in if-then form, the "if" part contains the hypothesis and the "then" part contains the conclusion.
Here is an example :
Conditional statements can be either true or false.
To show that a conditional statement is true, we must present an argument that the conclusion follows for all cases that fulfill the hypothesis.
To show that a conditional statement is false, describe a single counter example that shows the statement is not always true.
The converse of a conditional statement is formed by switching the hypothesis and conclusion.
Here is an example.
Statement : If you hear thunder, then you see lightning.
Converse : If you see lightning, then you hear thunder.
A statement can be altered by negation, that is, by writing the negative of the statement.
Here are some examples.
∠A = 65°
∠A is obtuse
∠A ≠ 65°
∠A is not obtuse
When we negate the hypothesis and conclusion of a conditional statement, we form the inverse. When we negate the hypothesis and conclusion of the converse of a conditional statement, we form the contrapositive.
If∠A = 65°, then ∠A is obtuse.
If∠A ≠ 65°, then ∠A is not obtuse.
If∠A is obtuse, ∠A = 65°.
If ∠A is not obtuse, then ∠A ≠ 65°.
In the above,
(i) Both Original and Contrapositive are true.
(ii) Both inverse and converse are false.
When two statements are both true or both false, they are called equivalent statements. A conditional statement is equivalent to its contrapositive. Similarly, the inverse and converse of any conditional statement are equivalent. This is shown above.
Example 1 :
Rewrite the following conditional statements in if-then form.
(i) Two points are collinear if they lie on the same line.
(ii) A number is divisible by 9 is also divisible by 3.
(iii) All sharks have a boneless skeleton.
(i) If two points lie on the same line, then they are collinear.
(ii) If a number is divisible by 9, then it is divisible by 3.
(iii) If a fish is a shark, then it would have a boneless skeleton.
Example 2 :
Write a counter example to show that the following conditional statement is false.
If x² = 25, then x = 5.
As a counter example, let us take x = -5.
The hypothesis is true, because (-5)² = 25. But, the conclusion is false, because it is given x = 5.
It implies that the given conditional statement is false.
Example 3 :
Write the converse of the following conditional statement.
"If two segments have the same length, then they are congruent"
"If two segments are congruent, then they have the same length"
Example 4 :
Write (a) inverse, (b) converse, (c) contrapositive of the following statement.
"If there is snow on the ground, the flowers are not in bloom"
(a) Inverse :
"If there is no snow on the ground, the flowers are in bloom"
(b) Converse :
"If flowers are not in bloom, then there is snow on the ground"
(b) Contrapositive :
"If flowers are in bloom, then there is no snow on the ground"
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