Lessons

1. Basics
2. Deductive Reasoning
3 - Parallel lines
4 - Congruent Triangles
6 - Inequalities
7 - Similar Polygon
8 - Rt. Triangles
9 - Circles
10 - Constructions
11 - Areas of 2D objects
12 - Areas and Volumes
13 - Coordinates
14 - Reflection/Rotation

If-Then Statements

Objective:

Understand If-Then Statements; Conditionals, Biconditional, Converse, Counterexample, Inverse, and Contrapositives.

• Know most of the properties from Algebra

Lesson 2-1 If-Then Statements:

If-Then Statements: An if-then statement is just what the name says it is. It is a statement that proves if something happens then something else will happen. For example,

“If Chris went to the store after school, then he will buy something.”

Or

“If D is between C and E, then CD + DE = CE”

These kinds of if-then statements are called conditional statements, or just conditionals.

To symbolize an if-then statement, then let p represent the hypothesis, and q represent the conclusion.

If p, then q.

# Converse: Switching the hypothesis and the conclusion forms a converse of a conditional.

Statement: If p, then q. Converse If q, then p.

A statement and a converse say different things. Some converses come out to be false while the statement is true. For example,

Statement: “If Chris lived in California, then he lives East of the capital of the USA”

Converse: “If Chris lives East of the capital of the USA, then he lives in California”

As anyone can see, the converse is false. If Chris lives east of the capital, there are a number of states he could live in, not just California. In cases such as these, where the hypothesis (p) is true and the converse (q) is false, there is a name. The names of these cases are called counterexample.

Conditional statesmen are not always written with the ‘if’ clause first. For example,

General Form

If p, then q

p implies q

p only if q

q if p

## Example

If x˛ = 4, then x = 2

x˛ = 4 implies x = 2

x˛ = 4only if x = 2

x = 2 if x˛ = 4

# Contrapostive: An inverse but switched around with the p and q. For example,

Statement: If p, then q

Inverse: If not p, then not q

Contrapositive: if not q, then not p

The statement is always true with the contrapositive, but a statement is not logically equivalent to its converse or to its inverse.

### Properties from Algebra:

Properties of Equality

 Addition Property If a = b and c = d, then a + c = b + d Subtraction Property If a = b and c = d, then a -c = b - d Multiplication Property If a = b, then ca = cb Division Property If a = b and c 0, the a/c = b/c Substitution Property If a = b, then either a or b may be substituted for the other in any equation. Reflexive Property a = a Symmetric Property If a = b, then b = a Transitive Property If a = b and b = c, then a = c
 Symmetric Property If segment DE = segment FG, the segment FG = segment DE or If

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