Classifying Triangles

Triangles get sorted two different ways, and every triangle has a name from each list. By sides: equilateral has three equal sides, isosceles has two, scalene has none. By angles: acute has all three angles under 90 degrees, right-angled has exactly one of 90, obtuse has one over 90. So describing something as a 'right-angled isosceles triangle' is perfectly normal — it's using one name from each list. And a useful consequence falls straight out: a triangle can have at most one right angle and at most one obtuse angle, because two of them would use up 180 degrees or more and leave nothing for the third.

Tear the three corners off a paper triangle and lay them side by side: they always form a straight line. That's the angle sum — a + b + c = 180 degrees — and it holds for every triangle regardless of shape or size. The practical consequence is that two angles always give you the third: subtract both from 180. Angles of 70 and 60 leave 50. That one fact solves most of the questions in this topic, and it's also what lets you check whether a set of angles is even possible — if three given angles don't add to 180, no such triangle exists.

The isosceles rule links sides to angles: equal sides sit opposite equal angles. So if a triangle has two equal sides and an apex angle of 40 degrees, the two base angles are equal — call them x — and 40 + x + x = 180 gives 2x = 140 and x = 70. The rule works both ways, which is just as useful: if you spot two equal angles, the triangle must be isosceles, and the sides opposite them are equal. An equilateral triangle then follows immediately: three equal sides means three equal angles, and 180 divided by 3 is 60. That's also why an equilateral triangle can never be right-angled.

Extend one side of a triangle and you create an exterior angle. It equals the sum of the two opposite interior angles — the ones it isn't touching. You can see why from two facts you already have: the exterior angle e and its neighbour b sit on a straight line, so e + b = 180; and the angle sum says a + b + c = 180. Setting those equal gives e = a + c. It's a handy shortcut, but it's worth knowing the derivation, because if you ever forget the rule you can rebuild it from the straight line and the angle sum in about ten seconds.

Geometry questions are marked on the reason as much as the answer. Writing 'x = 70' scores half of what's available even when it's right; writing '2x + 40 = 180 (angle sum of a triangle), so 2x = 140 and x = 70 (base angles of an isosceles triangle)' scores everything. The format is simple — one line of working, one reason in brackets — and there are only about six reasons you need in this topic: angle sum of a triangle, base angles of an isosceles triangle, angles on a straight line, exterior angle of a triangle, angles at a point, and vertically opposite angles. Learn those phrasings word for word and use them exactly.