Triangles

Three-sided polygons with rich properties connecting angles, sides, area, and similarity.

Triangle types by sides

Definition

A triangle is a closed shape with exactly three straight sides and three angles. The three angles always add up to exactly $180°$ .

Triangles are classified by their sides:

And by their angles:

In a right triangle, the longest side (opposite the right angle) is the hypotenuse. The Pythagorean theorem: $a^2 + b^2 = c^2$ .

Key properties

Angles always sum to $180°$ — no exceptions in flat (Euclidean) geometry
The longest side is always opposite the largest angle
Triangle inequality: each side must be shorter than the sum of the other two
An exterior angle equals the sum of the two non-adjacent interior angles

Angles in a triangle

A triangle has angles $50°$ and $70°$ . Find the third angle.

$\text{third angle} = 180° - 50° - 70° = 60°$

Common mistakes

Assuming right angle: not every triangle has a $90°$ angle — check before applying $a^2 + b^2 = c^2$
Triangle inequality: three lengths don't always form a triangle — e.g. $1$ , $2$ , $10$ cannot

Try it

An isosceles triangle has one angle of $40°$ at the apex. Find the two base angles.

Solution

The two base angles are equal. Let each be $x$ .

$40° + x + x = 180°$ , so $2x = 140°$ , giving $x = 70°$ .

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