Theorems: Derived Geometric Statements

Theorems are the cornerstone of mathematical proof. They are statements that have been proven true using a logical sequence of steps. This sequence relies on:

Definitions: Precise descriptions of mathematical terms.
Axioms (or Postulates): Statements assumed to be true without proof. They form the foundation of a mathematical system.
Previously Proved Theorems: Theorems that have already been established as true. They can be used as building blocks for proving new theorems.

Proving a theorem involves transforming a given set of assumptions (hypotheses) into a specific conclusion, using logical reasoning and the tools mentioned above. The process is rigorous and aims to leave no room for doubt.

Formulae

While the concept of a theorem doesn’t have its own specific formula, many theorems are expressed in terms of mathematical formulas. Formulas are often the conclusion of the theorem itself, or integral to its proof.

Examples

Here are two examples of theorems:

Example-1: The Pythagorean Theorem in a right-angled triangle.

Theorem: In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Formula: $a^2 + b^2 = c^2$, where ‘c’ is the hypotenuse, and ‘a’ and ‘b’ are the other two sides.

Example-2: The Triangle Sum Theorem.

Theorem: The sum of the interior angles of a triangle is always equal to 180 degrees.

Formula: $\angle A + \angle B + \angle C = 180^\circ$, where $\angle A$, $\angle B$, and $\angle C$ are the interior angles of a triangle.

Theorem with Proof

Let’s consider a proof of the Triangle Sum Theorem:

Theorem: The sum of the interior angles of a triangle is 180 degrees.

Proof:

Given: Triangle ABC.
Construction: Draw a line through vertex A parallel to side BC. Let this line be DE.
Angle Relationships:

$\angle DAB = \angle ABC$ (Alternate interior angles, because DE || BC).
$\angle EAC = \angle ACB$ (Alternate interior angles, because DE || BC).

Angle Sum: $\angle DAB + \angle BAC + \angle EAC = 180^\circ$ (Angles on a straight line).
Substitution: Substitute $\angle ABC$ for $\angle DAB$ and $\angle ACB$ for $\angle EAC$. This gives us: $\angle ABC + \angle BAC + \angle ACB = 180^\circ$.
Conclusion: Therefore, the sum of the interior angles of triangle ABC is 180 degrees. Q.E.D. (Quod Erat Demonstrandum – which was to be demonstrated).

Common mistakes by students

Students often struggle with theorems due to a few common mistakes:

Not understanding the definitions: A misunderstanding of definitions will impede any attempt to prove the theorem.
Lack of understanding the axioms: A student must be familiar with the basic rules
Assuming what they need to prove: Students sometimes make the mistake of incorporating the conclusion into their assumptions, which renders the proof invalid.
Misunderstanding the logical flow: Proving a theorem requires understanding that each step must logically follow from the previous steps. Students sometimes skip or misorder steps in their proofs.
Incorrect application of previously proven theorems: Applying a theorem in a way that isn’t valid for the given situation.

Real Life Application

Theorems are fundamental tools in various fields:

Engineering: Used in designing buildings, bridges, and other structures, relying on the properties of triangles, angles, and other geometric shapes.
Computer Science: Algorithms and program efficiency often rely on mathematical theorems for optimization and verification.
Cryptography: Secure communication relies on number theory theorems.
Finance: Used in modeling financial markets, risk assessment, and investment strategies.
Physics: Many physics laws are also expressed as theorems.

Fun Fact

Euclid’s “Elements,” written around 300 BC, is one of the most influential works in the history of mathematics. It presented geometry as a logical system, built on a foundation of axioms and theorems. It’s the foundational text on geometry which shaped how the field is taught for over two millennia!