Introduction to Proof

A mathematical proof is a logical argument that shows why a statement must be true. In geometry, proofs connect facts that are already known to a conclusion using definitions, properties, postulates, and theorems.

A proof does more than show that an answer appears to be correct. It explains why the answer must always be correct.

Mathematical Reasoning

Mathematical reasoning is the process of using information and logical relationships to reach a conclusion.

For example:

• All squares have four right angles.
• Shape ABCD is a square.
• Therefore, ABCD has four right angles.

The conclusion follows logically from the information given.

Geometry relies heavily on this kind of reasoning. Instead of accepting something because it looks true, we establish that it must be true.

Inductive Reasoning

Inductive reasoning uses observations or patterns to make a general prediction.

You examine several examples and use them to predict what will happen in other cases.

For example, suppose you investigate several rectangles:

• Rectangle 1 has an angle sum of 360°.
• Rectangle 2 has an angle sum of 360°.
• Rectangle 3 has an angle sum of 360°.

You might conclude:

The interior angles of every rectangle add to 360°.

This is a reasonable conclusion based on a pattern.

However, inductive reasoning does not prove that the conclusion must always be true.

Another Example

Consider the sequence:

2, 4, 6, 8, 10, ...

You observe that every number shown is even.

You might predict that the next number is 12.

This prediction is based on inductive reasoning.

Deductive Reasoning

Deductive reasoning begins with facts, definitions, or established mathematical rules and uses them to reach a logically certain conclusion.

For example:

• All right angles measure 90°.
• ∠ABC is a right angle.
• Therefore, ∠ABC = 90°.

If the starting information is true and the reasoning is valid, the conclusion must be true.

Deductive reasoning is the foundation of mathematical proof.

Inductive vs. Deductive Reasoning

Both are useful.

Inductive reasoning helps us discover ideas.

Deductive reasoning helps us prove them.

Conjectures

A conjecture is a mathematical statement that is believed to be true based on observations or patterns but has not yet been proven.

Suppose you draw several triangles and measure their interior angles.

You find:

Triangle 1: 60° + 70° + 50° = 180°

Triangle 2: 90° + 45° + 45° = 180°

Triangle 3: 30° + 80° + 70° = 180°

You might make the conjecture:

The interior angles of every triangle add to 180°.

Testing many triangles gives strong evidence, but measuring examples alone does not prove the statement for every possible triangle.

A deductive proof can establish that it is always true.

Statements and Reasons

Geometric proofs are often organized as a sequence of statements and reasons.

A statement tells us what is true.

A reason explains why we know the statement is true.

For example:

This structure makes the reasoning easy to follow.

Every important statement should have a valid reason.

Given Information

Most geometry proofs begin with information that the problem tells us is true.

This information is called the given.

For example:

Given: AB ∥ CD

This means you do not need to prove that AB and CD are parallel. You may use it as a starting fact.

Other givens might include:

• ∠A = 65°
• AB = AC
• AB ⟂ CD
• M is the midpoint of AB
• triangle ABC is isosceles
• ∠A and ∠B are vertically opposite

Always identify the given information before beginning a proof.

What Are You Trying to Prove?

A proof also has a conclusion or statement that must be demonstrated.

For example:

Given: AB ∥ CD

Prove: ∠1 = ∠2

Your job is to connect the given information to the required conclusion through a logical sequence.

Think of a proof as a chain:

Given information → known rules → intermediate conclusions → final conclusion

Every link in the chain must be justified.

Definitions in Proofs

Definitions are important because they tell us exactly what mathematical terms mean.

Midpoint

If M is the midpoint of AB, then:

AM = MB

Angle Bisector

If BD bisects ∠ABC, then:

∠ABD = ∠DBC

Perpendicular Lines

If AB ⟂ CD, then the lines form right angles.

Therefore, the angles at their intersection measure:

90°

Parallel Lines

Parallel lines can create equal corresponding and alternate angles when crossed by a transversal.

Definitions can therefore provide important steps in a proof.

Properties Used in Proofs

Several basic mathematical properties appear frequently in proofs.

Reflexive Property

A quantity is equal to itself.

AB = AB

This might seem obvious, but it becomes important when two shapes share a side.

Symmetric Property

If:

a = b

then:

b = a

Transitive Property

If:

a = b

and:

b = c

then:

a = c

Addition Property

If equal quantities have the same amount added to them, the results remain equal.

Subtraction Property

If equal quantities have the same amount subtracted from them, the results remain equal.

These properties allow us to build logical chains of reasoning.

Using Known Angle Theorems

Many simple geometric proofs use familiar angle relationships.

Important facts include:

• vertically opposite angles are equal
• angles on a straight line add to 180°
• angles around a point add to 360°
• corresponding angles are equal when lines are parallel
• alternate angles are equal when lines are parallel
• co-interior angles add to 180° when lines are parallel
• angles in a triangle add to 180°

These are not simply patterns we happen to notice. Once established mathematically, they can be used as reasons in other proofs.

A Simple Geometric Proof

Suppose two lines intersect. ∠1 and ∠2 are vertically opposite.

Given: ∠1 = 72°

Prove: ∠2 = 72°

The proof is short, but every step has a reason.

Proof Using Parallel Lines

Suppose lines AB and CD are parallel and a transversal crosses them.

Given: AB ∥ CD and ∠1 = 58°

Prove: ∠2 = 58°, where ∠1 and ∠2 are corresponding angles.

Notice that the statement AB ∥ CD is important. Without knowing the lines are parallel, we cannot automatically claim that the corresponding angles are equal.

A Multi-Step Proof

Suppose ∠A and ∠B form a straight line. ∠A = 65°, and ∠B and ∠C are vertically opposite.

Prove: ∠C = 115°.

First use the straight-line relationship:

∠A + ∠B = 180°

65° + ∠B = 180°

∠B = 115°

Then use vertically opposite angles:

∠B = ∠C

Therefore:

∠C = 115°

A statements-and-reasons version would be:

This is the basic structure of a longer proof: each conclusion provides information that can be used in the next step.

Proofs Involving Triangles

Triangles provide many opportunities for deductive reasoning.

Suppose triangle ABC has:

∠A = 50°

∠B = 60°

Find and justify ∠C.

We know:

∠A + ∠B + ∠C = 180°

because the interior angles of a triangle sum to 180°.

Therefore:

50° + 60° + ∠C = 180°

110° + ∠C = 180°

∠C = 70°

The important part of the reasoning is not merely the calculation. We also identify the theorem that makes the calculation valid.

Counterexamples

A counterexample is an example showing that a statement is not always true.

Suppose someone makes the conjecture:

All quadrilaterals have four equal sides.

A rectangle with unequal length and width is a counterexample.

Therefore, the conjecture is false.

This illustrates an important feature of mathematical reasoning:

One counterexample is enough to disprove a universal statement.

You might test 100 examples that support a conjecture, but they do not necessarily prove it. One valid counterexample can show that the conjecture is false.

Communicating a Proof Clearly

A good proof should be understandable to another person.

Use:

• correct mathematical vocabulary
• clearly identified givens
• statements in a logical order
• a reason for each important statement
• appropriate symbols
• calculations that are easy to follow
• a clear final conclusion

Avoid unexplained jumps.

For example, instead of:

x = 65°

write:

x = 65° because alternate angles between parallel lines are equal.

The second version communicates the mathematical reasoning.

A Strategy for Writing Proofs

When beginning a proof, ask:

What do I know?

Identify the givens.

What am I trying to prove?

Identify the final conclusion.

Which definitions or theorems connect them?

Look for relationships involving:

• parallel lines
• perpendicular lines
• angle bisectors
• midpoints
• equal lengths
• angle relationships
• triangles

Then build the argument one step at a time.

A useful mental structure is:

I know this → therefore this → which means this → therefore the conclusion follows.

Common Mistakes

Assuming something is true because it looks true

A diagram may not be drawn to scale. Use the given information and established geometric rules.

Confusing evidence with proof

Checking several examples can support a conjecture, but it does not necessarily prove the statement.

Giving statements without reasons

A proof must explain why each important conclusion follows.

Using a theorem when its conditions are not satisfied

For example, corresponding angles are equal when the lines are parallel. You cannot automatically use that relationship if parallel lines have not been established.

Repeating the conclusion instead of proving it

Writing "∠A = ∠B because they are equal" does not provide a mathematical reason.

Skipping steps

A reader should be able to follow the logical chain from the givens to the conclusion.

Key Terms

Proof — a logical argument demonstrating that a mathematical statement must be true.

Reasoning — the process of using information and logical relationships to reach a conclusion.

Inductive reasoning — using observations and patterns to form a general prediction or conjecture.

Deductive reasoning — using established facts, definitions, properties, and theorems to reach a logically certain conclusion.

Conjecture — a mathematical statement believed to be true based on evidence but not yet proven.

Counterexample — an example showing that a general statement is false.

Statement — a mathematical fact or conclusion appearing as a step in a proof.

Reason — the definition, property, theorem, or given information that justifies a statement.

Given — information supplied by a problem and accepted as true.

Conclusion — the statement that the proof is intended to establish.

Theorem — a mathematical statement that has been logically proven.

Definition — a precise explanation of the meaning of a mathematical term.

Two-column proof — a proof organized with statements in one column and the reasons for those statements in another.

Key Takeaways

• A proof explains why a mathematical conclusion must be true.
• Inductive reasoning uses examples and patterns to make conjectures.
• Deductive reasoning uses established facts and rules to reach certain conclusions.
• Examples can provide evidence for a conjecture, but examples alone usually do not prove a universal statement.
• One valid counterexample can disprove a universal statement.
• Geometric proofs often consist of statements and reasons.
• Definitions, properties, and previously established theorems provide reasons in proofs.
• Each step should follow logically from information already established.
• Diagrams should not be trusted simply because something looks equal, parallel, or perpendicular.
• Clear mathematical communication is an important part of proving a result.

Check Your Understanding

1. Explain the difference between inductive and deductive reasoning.
2. A student measures five triangles and finds that their interior angles each sum to 180°. Is this inductive or deductive reasoning? Explain.
3. A student knows that all right angles measure 90°. ∠A is a right angle, so the student concludes that ∠A = 90°. Is this inductive or deductive reasoning?
4. What is the difference between a statement and a reason in a geometric proof?
5. If M is the midpoint of AB, what statement can you make about AM and MB? Give the reason.
6. Lines AB and CD are parallel. Two corresponding angles are labelled x and y. What relationship exists between x and y, and why?
7. A student claims, "All quadrilaterals are rectangles." Give a counterexample.
8. ∠A and ∠B form a straight line. If ∠A = 47°, determine ∠B and justify your answer.
9. Two lines intersect. ∠1 = 125° and ∠1 and ∠3 are vertically opposite. Write a short statements-and-reasons proof showing that ∠3 = 125°.
10. Why is the statement "the angles look equal in the diagram" not an acceptable reason in a geometric proof?