Mathematical reasoning and proofs are a fundamental part of geometry. Several tools used in writing proofs will be covered, such as reasoning (inductive/deductive), conditional statements (converse/inverse/contrapositive), and congruence properties. The purpose of a proof is to prove that a mathematical statement is true.
Proof: A logical argument that uses logic, definitions, properties, and previously proven statements to show a statement is true.
Definition: A statement that describes a mathematical object and can be written as a biconditional statement.
Postulate: Basic rule that is assumed to be true. Also known as an axiom.
Theorem: Rule that is proven using postulates, definitions, and other proven theorems.
Congruent: When two geometric figures have the same shape and size.
Solving an algebraic equation is like doing an algebraic proof. Algebraic proofs use algebraic properties, such as the properties of equality and the distributive property.
Basic Algebraic Properties
Reflexive Property of Equality
a = a
Symmetric Property of Equality
a = b and b = a
Transitive Property of Equality
a = b and b = c, than a = c
Substitution Property of Equality
If a = b, then b can be used in place of a and vice versa
Addition Property of Equality
If a = b, then a + c = b + c
Subtraction Property of Equality
If a = b, then a - c = b - c
Multiplication Property of Equality
If a = b, then ac = bc
Division Property of Equality
If a = b, then a ÷ c = b ÷ c
a(b + c) = ab + ac
Although we may not write out the logical justification for each step in our work, there is an algebraic property that justifies each step.
- 3 = n + 1
-3 -1 = n +1 -1
Subtraction Property of Eguality
-4 = n
n = -4
Symmetric Property of Equality.
Since segment lengths and angle measures are real numbers, the following properties of equality are true for segment lengths and angle measures:
A proof is a logical argument that shows a statement is true.
Definitions, postulates, properties, and theorems can be used to justify each step of a proof.
Be careful when interpreting diagrams. There are some things you can conclude and some that you cannot.
• Collinear points
• Measures of segments
• Coplanar points
• Measures of angles
• Straight angles and lines
• Congruent segments
• Adjacent angles
• Congruent angles
• Linear pairs of angles
• Right angles
• Vertical angles
A direct geometric proof is a proof where you use deductive reasoning to make logical steps from the hypothesis to the conclusion. Each logical step needs to be justified with a reason.
There are several types of direct proofs:
A two-column proof is one way to write a geometric proof. Starting from GIVEN information, use deductive reasoning to reach the conjecture you want to PROVE.
Example of a Two-Column Proof:
An indirect proof is where we prove a statement by first assuming that it’s false and then proving that it’s impossible for the statement to be false (usually because it would lead to a contradiction). If the statement cannot be false, then it must be true.
Steps to write an indirect proof:
Use variables instead of specific examples so that the contradiction can be generalized.