Triangle Basics

A triangle has three vertices, three sides and three angles. In $\triangle ABC$, the angles are $A$, $B$ and $C$, and by convention the side opposite each angle uses the lowercase letter: side $a$ is across from angle $A$.

Angle sum

The three angles of every triangle add up to $180°$:

$$A + B + C = 180°$$

Draw a line through one vertex parallel to the opposite side and you'll see the three angles line up along it. If two angles are $50°$ and $65°$, the third is $180° - 115° = 65°$.

Exterior angles

Extend one side past a vertex. The exterior angle formed equals the sum of the two remote interior angles. With $A = 40°$ and $B = 75°$, the exterior angle at $C$ is $115°$, because both it and $A + B$ make $180°$ with angle $C$.

Sides and angles

• The largest angle is opposite the longest side, and the smallest is opposite the shortest.
• Triangle inequality: each side is shorter than the other two combined, $a < b + c$. So $3, 4, 5$ makes a triangle, but $2, 3, 6$ doesn't, since $2 + 3 < 6$ and the short sides can't reach.

If two sides are $7$ and $10$, the third side must be between $10 - 7 = 3$ and $10 + 7 = 17$.

Area

$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$

The height is measured perpendicular to the base, and it may fall outside the triangle when one angle is obtuse. Section 03.11 gives a formula that uses an angle instead of the height.