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If-Then Statements An if-then statement is a statement such as “If you are
Converse, Inverse, and Contrapositive If you change the hypothesis or
reading this page, then you are studying math.” A statement that can be
written in if-then form is called a conditional statement. The phrase
immediately following the word if is the hypothesis. The phrase immediately
following the word then is the conclusion.
conclusion of a conditional statement, you form related conditionals. This
chart shows the three related conditionals, converse, inverse, and
contrapositive, and how they are related to a conditional statement.
A conditional statement can be represented in symbols as p → q, which is
read “p implies q” or “if p, then q.”
Example 1: Identify the hypothesis and conclusion of the conditional
Symbols
Example
Conditional
p→q
If two angles are vertical angles,
then they are congruent.
Converse
q→p
Inverse
∽p → ∽q
Contrapositive
∽q → ∽p
statement.
If ∠𝑋 ≅ ∠𝑅 and ∠𝑅 ≅ ∠𝑆 , then ∠𝑋 ≅ ∠𝑆.
Example 2: Identify the hypothesis and conclusion. Write the statement
in if-then form.
You receive a free pizza with 12 coupons.
Exercises
Identify the hypothesis and conclusion of each conditional statement.
1. If it is Saturday, then there is no school.
2. If x – 8 = 32, then x = 40.
Write each statement in if-then form.
3. All apes love bananas.
If the conditional is true…. Then the _________________ is also always true.
Exercises
Write the converse, inverse, and contrapositive of each true conditional
statement. Determine whether each related conditional is true or false. If
a statement is false, find a counterexample.
1. If you live in San Diego, then you live in California.
4. The sum of the measures of complementary angles is 90.
5. Collinear points lie on the same line.
2. If two angles are complementary, then the sum of their measures is 90.
Determine the truth value of each conditional statement.
6. If a is an integer, then 10a is greater than a.
7. If an odd number and an even number are multiplied, then the product is
even.
2-3 Skills Practice
Identify the hypothesis and conclusion of each conditional statement.
1. If you purchase a computer and do not like it, then you can return it
within 30 days.
2. If x + 8 = 4, then x = –4.
3. If the drama class raises $2000, then they will go on tour.
4. If 3x + 4 = –5, then x = –3.
11. If I roll two six-sided dice and sum of the numbers is 11, then one die
must be a five.
12. If two angles are supplementary, then one of the angles is acute.
Write the converse, inverse, and contrapositive of the conditional
statement.
Determine whether each statement is true or false. If a statement is
false, find a counterexample.
13. If a number is divisible by 2, then the number is even.
5. If you take a class in television broadcasting, then you will film a sporting
event.
14. If a and b are negative, then a + b is also negative.
Write each statement in if-then form.
6. A polygon with four sides is a quadrilateral.
7. An acute angle has a measure less than 90.
15. If two triangles have equivalent angle measures, then they are
congruent.
8. “Those who do not remember the past are condemned to repeat it.”
(George Santayana)
9. Adjacent angles share a common vertex and a common side.
16. Older campers who attend Camp Longhorn wait on tables.
a. Write a conditional statement in if-then form.
Determine the truth value of each conditional statement.
If false, give a counterexample.
10. If you have five dollars, then you have five one-dollar bills.
b. Write the converse of your conditional statement.