Honors Geometry Unit 2 Review: Logic and Proof

An algebraic proof establishes an algebraic result, such as a solution to an equation, with a named justification for each step. The algebra is usually easy. The skill is citing the right property.

The properties of equality do most of the work. If $a = b$, you can add, subtract, or multiply both sides by the same number $c$. You can also divide both sides by $c$ when $c \neq 0$. The Reflexive Property says $a = a$, the Symmetric Property says if $a = b$ then $b = a$, and the Transitive Property says if $a = b$ and $b = c$ then $a = c$. Substitution lets you swap equal quantities, and the Distributive Property justifies $2(x + 3) = 2x + 6$ and combining like terms.

Geometry enters when a definition turns a figure into an equation. A midpoint gives two equal lengths, and supplementary angles give a sum of 180°.

Worked example. Given that $M$ is the midpoint of $\overline{AB}$, $AM = 2x + 5$, and $MB = 4x - 7$, prove $x = 6$.

1. $AM = MB$ (Definition of midpoint)
2. $2x + 5 = 4x - 7$ (Substitution Property)
3. $5 = 2x - 7$ (Subtraction Property of Equality, subtracting $2x$)
4. $12 = 2x$ (Addition Property of Equality)
5. $6 = x$ (Division Property of Equality)
6. $x = 6$ (Symmetric Property of Equality)

Plugging back in gives $AM = 17$ and $MB = 17$, which checks out.

The three proof formats hold the same logic. They only differ in layout.

A two-column proof lists statements on the left and reasons on the right. It starts with the given information and ends with the exact statement you were asked to prove. Every row needs a reason, even the steps that feel obvious. This is the format most honors classes use first, because it's easy to check line by line.

A paragraph proof says the same steps in full sentences, with the reasons built in. A typical sentence reads "Since $M$ is the midpoint of $\overline{AB}$, $\overline{AM} \cong \overline{MB}$ by the definition of midpoint." Words like since, because, and therefore carry the logic. Paragraph proofs read naturally, but long ones can hide a missing step.

A flow proof puts each statement in a box and draws arrows to show what leads to what, with the reason written under each box or arrow. Flow proofs can branch and merge. If a conclusion needs two earlier facts, two arrows point into its box, so you can see the structure at a glance.

The reasons you're allowed to cite are the same in every format. They are given information, definitions, postulates, theorems, and algebraic properties. "It looks that way in the diagram" is never a valid reason.

A conditional statement has the form "If $p$, then $q$," written $p \rightarrow q$. The part after "if" is the hypothesis, and the part after "then" is the conclusion. The order in the sentence can change. In "A triangle is isosceles if it has two congruent sides," the hypothesis is still "it has two congruent sides."

You can rearrange a conditional three ways. The converse swaps the parts ($q \rightarrow p$). The inverse negates both ($\sim p \rightarrow \sim q$). The contrapositive swaps and negates ($\sim q \rightarrow \sim p$).

Two statements are logically equivalent if they always have the same truth value. A conditional and its contrapositive are always equivalent. The converse and inverse are equivalent to each other, but they are not necessarily equivalent to the original. A conditional is false only when the hypothesis is true and the conclusion is false, so one counterexample proves it false.

Worked example. Take "If $x^2 = 25$, then $x = 5$." Try $x = -5$. Then $x^2 = 25$ but $x \neq 5$, so the original is false. The contrapositive, "If $x \neq 5$, then $x^2 \neq 25$," fails for the same $x = -5$, so it's also false. The converse, "If $x = 5$, then $x^2 = 25$," is true, and so is the inverse, "If $x^2 \neq 25$, then $x \neq 5$."

When a conditional and its converse are both true, you can join them into a biconditional with "if and only if" ($p \leftrightarrow q$). Every good definition works this way.

Inductive reasoning goes from specific examples to a general rule. You notice a pattern and make a conjecture, which is an educated guess. Deductive reasoning uses facts, including definitions, postulates, and theorems, to reach a conclusion that must be true if the facts are true and the logic is valid. The conclusion can be specific or general. Induction finds ideas, and deduction proves them.

Two deductive rules show up constantly. The Law of Detachment says that if $p \rightarrow q$ is true and $p$ is true, then $q$ is true. The Law of Syllogism says that if $p \rightarrow q$ and $q \rightarrow r$ are both true, then $p \rightarrow r$ is true. A related rule, sometimes called modus tollens, says that if $p \rightarrow q$ is true and $q$ is false, then $p$ is false.

Checking examples does not prove a conjecture unless you check every possible case. One counterexample is enough to disprove it.

Worked example. Test the conjecture "$n^2 + n + 41$ is prime for every whole number $n$." For $n = 0$ you get $41$. For $n = 1$ you get $43$, for $n = 2$ you get $47$, and for $n = 3$ you get $53$. All are prime, and the pattern keeps working for a long time. Now try $n = 40$. You get $1600 + 40 + 41 = 1681 = 41 \times 41$, which is not prime. Forty straight successes still didn't prove the rule, and one failure ended it.