Geometry Unit 2: Reasoning and Proof | Notes and practice

Notes

Conditional statements

Statement	Form
Conditional	$p \to q$
Converse	$q \to p$
Inverse	not $p \to$ not $q$
Contrapositive	not $q \to$ not $p$

A conditional and its contrapositive have the same truth value. So do the converse and inverse.
A biconditional ($p$ if and only if $q$) is true when the conditional and converse are both true. Good definitions are biconditionals.
One counterexample shows a conditional is false.

Reasoning

Inductive: pattern to conjecture (may be wrong). Deductive: facts and logic to a certain conclusion.
Detachment: $p \to q$ and $p$ give $q$. Syllogism: $p \to q$ and $q \to r$ give $p \to r$.

Proof tools

Properties of equality: addition, subtraction, multiplication, division, substitution, distributive.
Reflexive, symmetric and transitive properties work for both equality and congruence.
A two-column proof starts with the given, and every statement needs a reason.
Key theorems: vertical angles are congruent, linear pairs are supplementary, all right angles are congruent, congruent supplements (or complements) are congruent.

Topics in this unit

Each topic has its own short notes and practice.

Practice

What is the converse of "If a triangle is equilateral, then it is isosceles"?

A conditional statement is true. Which related statement must also be true?

"If two angles form a linear pair, then they are supplementary." $\angle 1$ and $\angle 2$ form a linear pair. Which law lets you conclude they are supplementary?

Which reason justifies: if $m\angle 1 + m\angle 2 = 90^\circ$ and $m\angle 1 = 30^\circ$, then $30^\circ + m\angle 2 = 90^\circ$?

$\angle 1$ and $\angle 2$ are complementary, and $\angle 3$ and $\angle 2$ are complementary. If $m\angle 1 = (2x + 10)^\circ$ and $m\angle 3 = (4x - 8)^\circ$, find $m\angle 1$.

A student notices $1 + 3 = 4$, $3 + 5 = 8$ and $5 + 7 = 12$, and concludes "the sum of two odd numbers is even." What kind of reasoning is this?

Free response

Write your answer first, then open the scoring guide.

Free-response question

A school's dress code says: "If it is a game day, then students may wear team jerseys."

(a) Write the converse and decide whether it must be true. Explain.

Show the answer and scoring guide

Converse: "If students may wear team jerseys, then it is a game day." It is not necessarily true; the school could also allow jerseys on a spirit day, for example.

Check off the points you earned:

1 point: correct converse
1 point: says it is not necessarily true, with a reason or counterexample

(b) Write the contrapositive. Is it true whenever the original rule is true?

Show the answer and scoring guide

Contrapositive: "If students may not wear team jerseys, then it is not a game day." Yes; a conditional and its contrapositive are always both true or both false.

Check off the points you earned:

1 point: correct contrapositive
1 point: yes, with the equivalence reason

(c) Given: $m\angle A + m\angle B = 180^\circ$ and $m\angle A = 3x$ and $m\angle B = 2x + 30$. Give a short two-column proof that $x = 30$, with a reason for each step.

Show the answer and scoring guide

$m\angle A + m\angle B = 180^\circ$ (Given). $3x + 2x + 30 = 180$ (Substitution). $5x + 30 = 180$ (Combine like terms / Simplify). $5x = 150$ (Subtraction Property). $x = 30$ (Division Property).

Check off the points you earned:

1 point: correct algebra steps in order ending with $x = 30$
1 point: a valid reason for every step