# Three Types of Geometric Proofs You Need to Know

**Three Types of Geometric Proofs You Need to Know**

Geometry is an essential branch of mathematics that studies objects’ shapes, sizes, and positions in space. One of the core components of geometry is the art of proof, where students learn to demonstrate the validity of a statement or theorem through logical reasoning. In this article, we’ll explore three types of geometric proofs that every student needs to know, including **two-column proof**.

Table of Contents

**What is a Two-Column Proof?**

A **two-column proof in geometry** is structured with two columns, where one presents the logical steps taken. At the same time, the other provides a corresponding reason or justification for each step. This type of proof is a powerful tool for organizing geometric reasoning and demonstrating the logical sequence of statements used to prove a theorem or solve a problem.

**The three types of geometric proofs we will discuss include the following:**

- Direct Proof
- Indirect Proof
- Proof by Contradiction

**Direct Proof**

Direct proof is a type where the conclusion follows directly from the premises. In other words, the argument proceeds straightforwardly, starting from a set of factual statements and ending with the desired result. Direct proof is the most common type used in geometry and is often used to prove theorems about congruent triangles, parallel lines, and similar figures.

**Example:**

Prove that if two angles are complementary, their sum is 90 degrees.

Statement 1: ∠A and ∠B are complementary angles

Statement 2: m∠A + m∠B = 90° (definition of complementary angles)

Conclusion: The sum of two complementary angles is 90 degrees.

**Indirect Proof**

The indirect proof is a type of proof where we assume the opposite of what we want to prove and then show that this assumption leads to a contradiction. The indirect evidence often proves theorems involving perpendicular lines, congruent triangles, and similarity.

**Example:**

Prove that there is no smallest rational number.

Assumption: Let a be the smallest rational number.

Statement 1: a = p/q, where p and q are coprime integers.

Statement 2: b = a/2 is also a rational number.

Statement 3: b < a (since a is the smallest rational number).

Statement 4: b = m/n, where m and n are coprime integers.

Statement 5: p > m (since b < a).

Statement 6: p/q > m/n (by substitution of statements 1 and 4)

Conclusion: Since p > m and p and m are coprime, this contradicts the assumption that p and q are coprime.

**Proof by Contradiction**

It is a type of proof where we assume the opposite of what we want to prove and then show that this assumption leads to a contradiction. The critical difference between proof by polarity and indirect proof is that proof by contradiction assumes the opposite of the conclusion we want to prove. In contrast, indirect proof takes the opposite of a premise.

Example:

Prove that the square root of 2 is irrational.

Assumption: Let √2 be a rational number.

Statement 1: √2 = p/q, where p and q are coprime integers.

Statement 2: 2 = p²/q² (by squaring both sides of statement 1).

Statement 3: p² = 2q².

Statement 4: p is even (since the square of an odd number is odd, and the court of an even number is even).

Statement 5: q is even (since p² is divisible by 2, and p is even).

Conclusion: Since p and q are both even, this contradicts the assumption that p and q are coprime. Therefore, the assumption that √2 is rational must be false, and it follows that √2 is irrational.

Finally, direct proof, indirect proof, and proof by contradiction are three fundamental types of two column proof that every student needs to know. Every kind of proof has its unique structure and can be used to prove theorems and solve problems in different contexts.

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