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Triangles & Proofs

The triangle is geometry's most fundamental shape — three sides, three angles, infinite depth.

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Triangle Classification

Triangles can be classified by their sides (equilateral, isosceles, scalene) or angles (acute, right, obtuse). Every triangle has an angle sum of 180° — a fact that's specific to Euclidean geometry and fails in non-Euclidean geometry.

The Angle Sum Property

In any triangle: A + B + C = 180°

This can be proved by drawing a line through one vertex parallel to the opposite side and using alternate interior angles. It's one of the first results students prove in geometry, and it leads directly to exterior angle theorems and polygon angle formulas.

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Congruence

Two triangles are congruent (identical in shape and size) if they satisfy one of these conditions:

  • SSS: Three pairs of sides are equal
  • SAS: Two sides and the included angle are equal
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  • ASA: Two angles and the included side are equal
  • AAS: Two angles and a non-included side are equal
Note: SSA (two sides and a non-included angle) does NOT guarantee congruence — this is the famous "ambiguous case" that also appears in the Law of Sines.

Similarity

Similar triangles have the same shape but possibly different sizes. Their corresponding angles are equal and corresponding sides are proportional:

△ABC ~ △DEF ⟹ AB/DE = BC/EF = AC/DF

Conditions: AA (two angles equal), SAS~ (proportional sides with equal included angle), SSS~ (all sides proportional).

Similarity is the basis of trigonometry — the trig ratios (sin, cos, tan) are well-defined precisely because all right triangles with the same acute angle are similar.

The Pythagorean Theorem

a² + b² = c²   (where c is the hypotenuse)

This is arguably the most famous theorem in mathematics. It connects geometry to algebra, enables the distance formula in coordinate geometry, and generalizes to the law of cosines for non-right triangles.

Example: Find c when a = 5, b = 12

c² = 25 + 144 = 169 → c = 13

(5, 12, 13) is a Pythagorean triple — all integers!

Pythagorean triples (integer solutions to a² + b² = c²) connect to number theory. Fermat's Last Theorem — proved by Andrew Wiles in 1995 — states that aⁿ + bⁿ = cⁿ has no positive integer solutions for n > 2.

Geometric Proof Techniques

Two-Column Proofs

The classic format: left column for statements, right column for reasons (given, definition, theorem, etc.).

Proof by Contradiction

Assume the opposite, derive a contradiction. Example: prove √2 is irrational (a result from number theory that has deep geometric meaning — it's the diagonal of a unit square).

Coordinate Proofs

Place figures on the coordinate plane and use algebra to prove geometric results. This bridges geometry and algebra powerfully.

A deep theme in mathematics is the interplay between geometry and algebra. Coordinate geometry translates geometric problems into equations. Linear algebra generalizes these ideas to any number of dimensions. This interplay is at the heart of modern mathematics.

Related Topics

The Unit Circle Right triangle trig ratios extended to the unit circle. Coordinate Geometry Use algebra to prove geometric properties and find distances. Prime Numbers Pythagorean triples and irrational numbers connect to number theory. Vectors & Matrices Generalize geometric operations to higher dimensions.