Angle Sums of Triangles and Quadrilaterals

The angles of any triangle add to 180 degrees. You can see why by tearing the three corners off a paper triangle and lining them up — they always form a straight line, which is 180. This holds for every triangle, however big, small, thin or stretched. So if you know two angles, subtract both from 180 to get the third: 55 and 65 leave 180 minus 120, which is 60. Write the reason as 'angle sum of a triangle'. A useful consequence: a triangle cannot have two right angles or two obtuse angles, because two 90s already use the whole 180. Every triangle has at least two acute angles.

The angles of any quadrilateral add to 360 degrees, and the reason is neat: draw one diagonal and it splits into two triangles, each worth 180, so the total is 360. If three angles are known, subtract all three from 360 — with 100, 85 and 95, that is 360 minus 280, which is 80. You can check the rule against shapes you already know: a rectangle has four right angles, and 4 times 90 is 360 ✓. Same for a square. Write your reason as 'angle sum of a quadrilateral'.

Special triangles hand you extra information, because equal sides always come with equal angles. An isosceles triangle has two equal sides and therefore two equal base angles. If the odd angle at the top is 40, the other two share 180 minus 40, which is 140, so each is 70. Subtract the odd angle from 180, then halve what is left. An equilateral triangle has three equal sides, so all three angles are equal at 180 divided by 3, which is 60 — always 60, no calculation needed. Look for tick marks on the sides in a diagram, because they tell you which sides are equal and therefore which angles match.

Extend one side of a triangle and you create an exterior angle, which comes with a shortcut. The long way is to find the third interior angle first, then take it from 180. The shortcut is that the exterior angle equals the sum of the two opposite interior angles — with 70 and 50 inside, the exterior angle is simply 120. Same answer, one step instead of two. The word opposite matters: it means the two angles the exterior angle does not sit next to. The trap is including the interior angle right beside it, which is supplementary to the exterior angle rather than part of the sum. Sense-check by confirming the exterior angle is bigger than either angle you added.

Harder questions are angle chases, where you cannot reach the answer in one move. The method is to work outwards from what you know. Mark every given angle on the diagram. Find any angle you can in a single move and write it on. Repeat, because each new angle unlocks more. Give a reason at every step, and finish by checking your totals — triangle 180, quadrilateral 360, angles at a point 360. The mistake is hunting for a clever route straight to the answer. Do not. Find any angle you can, write it on the diagram, and look again — hard problems fall apart once two or three extra angles are filled in.