Geometric Proofs

Formal chains of logical deduction that establish geometric truths from axioms, definitions, and previously proven theorems.

A proof turns a visual claim into a chain of necessary reasons

Reason chain

angle 1 + angle 2 = 180

angle 2 + angle 3 = 180

Therefore angle 1 = angle 3.

Definition

A geometric proof is a logical argument that shows why a geometric statement must be true. It is not enough to observe that something seems true or to check a few examples — a proof shows it is always true, for every possible case, forever.

A proof consists of:

Given information: what you are told is true
Steps: each step follows from the previous ones using definitions, axioms, or previously proven theorems
Conclusion: what you have shown to be true

Proofs can be written as two-column tables (statements in one column, reasons in the other) or as flowing paragraphs. The format doesn't matter; the logic does.

Proving vertical angles are equal

Given: Lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ intersect at point $P$ , forming angles $\angle 1$ , $\angle 2$ , $\angle 3$ , $\angle 4$ in order around $P$ .

Prove: $\angle 1 = \angle 3$ (vertical angles are equal).

Proof: $\angle 1$ and $\angle 2$ are supplementary (they form a straight line), so $\angle 1 + \angle 2 = 180°$ .

Similarly, $\angle 2 + \angle 3 = 180°$ .

Therefore $\angle 1 + \angle 2 = \angle 2 + \angle 3$ , which gives $\angle 1 = \angle 3$ . $\square$

Try it

Prove that the sum of the angles in a triangle is $180°$ .

Solution

Given: Triangle $ABC$ .

Prove: $\angle A + \angle B + \angle C = 180°$ .

Proof: Draw a line through $B$ parallel to $AC$ .

Label the angles at $B$ : the angle to the left of $\angle B$ as $\angle 1$ and to the right as $\angle 2$ .

Since the line through $B$ is parallel to $AC$ :

$\angle 1 = \angle A$ (alternate interior angles)
$\angle 2 = \angle C$ (alternate interior angles)

The three angles at $B$ form a straight line: $\angle 1 + \angle B + \angle 2 = 180°$ .

Substituting: $\angle A + \angle B + \angle C = 180°$ . $\square$

Related concepts

Needs first

Angles Triangles

Points, Lines & Planes Similarity

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