Types of Statements
In the conditional statement “If it rains, then the ground is wet,” what is the hypothesis?
If it rains
This type of reasoning uses patterns and observations to form a conclusion.
Inductive
This symbol, ∪, represents the operation that combines all elements from two sets.
In a two-column proof, these two parts make up the columns.
Statements and Reasons
Two angles whose measures sum to
90∘
.
Complementary
This type of statement switches the hypothesis and conclusion of a conditional statement.
Converse
This type of reasoning uses facts, definitions, and logic to reach a conclusion.
Deductive
This symbol, ∩, represents the operation that finds only the common elements between two sets.
Intersection
1. 2x= 4
2. x=2
What is the reasoning to get from step 1 to 2?
Division Property of Equality
Two angles whose measures sum to
180∘
.
Supplementary
Given the statement “If a figure is a square, then it has four sides,” this type statement would be “If a figure is not a square, then it does not have four sides.
Inverse
A student notices that
2+2=4
4+4=8 and
6+6=12
then concludes that doubling a number always gives an even result. This is an example of this type of reasoning.
Inductive
If Set A = 1,2,3,4
and Set B = 3,4,5,6
What is A∩B
{3,4}
2(x+5)=3(x+2)
Complete the first step. What is the reasoning for the step?
2x+10=3x+6
Definition of Distributive Property
When two lines intersect, these pairs of opposite angles are always congruent.
Vertical
This is the only related conditional statement that is logically equivalent to the original conditional.
Contrapositive
An example that proves a conjecture false.
Counterexample
In a Venn diagram with two overlapping circles, elements that belong to both sets are placed in this region.
Intersection, overlapping
2 angles add to 90 degrees. What is the reasoning you could use for this?
Definition of Complementary angles
When a transversal crosses two parallel lines, these angles are in corresponding positions and are congruent.
Corresponding Angles
Given “If x=3, then x2=9,” write the contrapositive statement.
If x squared does not = 9, then x does not equal 3
Given: All linear pairs are supplementary. Angles A and B form a linear pair. Using deductive reasoning, what can you conclude about angles A and B?
Angles A and B are supplementary,” or “their measures sum to 180∘
If Set A has 8 elements, Set B has 6 elements, and their intersection has 3 elements, this is the number of elements in A∪B
11
Write a full proof for the following:
x+2x+3=9
x+2x+3=9 Given
3x+3=9 Combine Like Terms
3x=6 Subtraction Property of Equality
x=2 Division Property of Equality
2 angles are supplementary. One of the angles is 142 degrees. What is the other angle?
38