🔺 Triangles

• Angle sum: the three interior angles always add to $180^\circ$, i.e. $\angle A + \angle B + \angle C = 180^\circ$. So a missing angle is just $180^\circ$ minus the other two.

• Exterior angle: an exterior angle equals the sum of the two opposite (non-adjacent) interior angles. This skips finding the third interior angle. Beware the trap: the angle next to the exterior angle is its supplement ($180^\circ - \text{exterior}$), not the answer.

• Angle ratio: if the angles are in ratio $a:b:c$, write them as $ax, bx, cx$, set $ax+bx+cx = 180^\circ$, solve for $x$, then read the part asked.

• Triangle inequality: the sum of any two sides is strictly greater than the third. For a third side $x$ with given sides $a$ and $b$, this becomes the open range $(a-b) < x < (a+b)$ — never $\le$ or $\ge$, because equality flattens the triangle into a line.

• Isosceles: equal sides face equal angles (the base angles are equal). An equilateral triangle is the special case — all sides equal, every angle $60^\circ$.

• Pythagoras: in a right triangle $a^2 + b^2 = c^2$ (with $c$ the hypotenuse). Memorise the triplets $3\text{-}4\text{-}5$, $5\text{-}12\text{-}13$, $8\text{-}15\text{-}17$ (and their multiples) to skip the arithmetic.

• Area: for any triangle, area $= \tfrac12 \times \text{base} \times \text{height}$. Equating two ways of writing the area gives the altitude to the hypotenuse: $h = \dfrac{\text{leg}\times\text{leg}}{\text{hyp}}$.