Rohit Thaiparambil - Geom 1.3
Lesson 5.4 Study Guide
This section is all about the opposite stuff. There are negative statements all over it. This chart will basically demonstrate what each key term is.
Statement
Conditional
Negation
Inverse
Contrapositive
(converse of the inverse)
Example
If a triangle has 3 equal sides, then it is equilateral.
A triangle does not have 3 equal sides.
If a triangle does not have 3 equal sides, then it is not equilateral.
If a triangle is not equilateral, then it does not have 3 equal sides.
In English
If a, then b
Not a
If not a, then not b
(Basically negate both parts of the conditional)
If not b, then not a
(Switch the statements from the inverse)
A conditional and its contrapositive statement are equivalent statements, meaning they have the same truth value (both tell the same thing).
A conditional and its inverse statement can have either the same or different truth values.

Statement: The baseball game is on Wednesday
Negation: The baseball game is not on Wednesday
Statement: The TV is not on
Negation: The TV is on
Statement: If you work out every day, you will be healthier.
Inverse: If you don't work out every day, you will not be healthier.
Statement: If a quadrilateral has 4 equal sides, then it is a square.
Inverse: If a quadrilateral doesn't have 4 equal sides, then it is not a square.
Statement: If a triangle has exactly 2 equal sides, then it is isosceles.
Contrapositive: If a triangle is not isosceles, then it does not have exactly 2 equal sides.
Statement: If the temperature is below 32 degrees Fahrenheit, then rain will freeze.
Contrapositive: If rain does not freeze, then the temperature is not below 32 degrees Fahrenheit

(Prove a statement true by proving its negation false through contradictions)
Step 1: Write the negation of what you want to prove. Put
"assume that" before the negation.
Step 2: Explain how if the assumption was true, it would contradict something such as a postulate or theorem.
Step 3: Conclude that the assumption (the opposite of the
Prove statement) is false and that the original Prove statement is true.

Given: The sum of the angle measures of triangle ABC is 180.
Prove: Triangle ABC cannot have 3 right angles.
Step 1: Assume that triangle ABC can have 3 right angles
Step 2: A right angle has a measure of 90 degrees. This means that the sum of the angle measures is 270. This contradicts the given statement that the sum of the angle measures of triangle ABC is 180.
Step 3: Now you can conclude that the assumption is false. Therefore, the original Prove statement saying triangle ABC cannot have 3 right angles is true.
B
A C

Negate the following statements:
1.The tree is very old.
2. The triangle is scalene.
Find the inverse of the following conditionals:
3. If you eat all you veggies, you will grow.
4. If a figure is a triangle, then it has exactly 3 sides.
Find the contrapositive of the following conditionals:
5. If a leaf is green, then it reflects green light.
6. If a shape has 2 pairs of parallel sides, then it is a parallelogram.
Indirect Proof:
7. Given: The cost of 2 items is $30
Prove: At least one of the items cost $15 or more.