Page 291 - ABCTE Study Guide_Neat

This would be the general form of many types of logical statements that would be similar to: "if Joe has 5
cents, then Joe has a nickel or Joe has 5 pennies ". Basically, a proof is a flow of implications starting with
the statement p and ending with the statement q. The stepping stones we use to link these statements in
a logical proof on the way are called axioms or postulates, which are accepted logical tools.

A direct proof will attempt to lay out the shortest number of steps between p and q.

The goal of an indirect proof is exactly the same--it wants to show that q follows from p; however, it goes
about it in a different manner. An indirect proof also goes by the names "proof by contradiction"
or reductio ad absurdum. This type of proof assumes that the opposite of what you want to prove is
true, and then shows that this is untenable or absurd, so, in fact, your original statement must be true.

Let's see how this works using the isosceles triangle below. The indirect proof assumption is in bold.

Given: Triangle ABC is isosceles with B marking the vertex.

Prove: Angles A and C are congruent.

Now, let's work through this, matching our statements with our reasons.

1. Triangle ABC is isosceles . . . . . . . . . . . . Given

2. Angle A is the vertex . . . . . . . . . . . . . . . . Given

3. Angles A and C are not congruent . . Indirect proof assumption

4. Line AB is equal to line BC . . . . . . . . . . .Legs of an isosceles triangle are congruent

5. Angles A and C are congruent . . . . . . . . The angles opposite congruent sides of a triangle are
congruent

6. Contradiction . . . . . . . . . . . . . . . . . . . . . .Angles can't be congruent and incongruent

7. Angles A and C are indeed congruent . . . The indirect proof assumption (step 3) is wrong

8. Therefore, if angles A and C are not incongruent, they are congruent.

“Always, Sometimes, and Never”

Some math problems work on the mechanics that statements are “always”, “sometimes” and “never” true.