A triangle is the simplest shape you can draw with straight sides — and yet it hides one of the most reliable facts in all of mathematics. No matter how you stretch it, squash it, or spin it, its three angles always add up to exactly the same number. Once you trust that, triangles stop being random and start telling you their secrets.
In this lesson you'll see why the angles always sum to \(180°\), learn the simple vocabulary for naming triangles, and then meet a genuinely useful idea: how to tell when two triangles are exactly the same — without measuring every single part.
The 180° rule
Here is the headline fact, and it is worth memorising: the three interior angles of any triangle add up to \(180°\). A flat, scalene, wonky triangle? \(180°\). A tall thin one? \(180°\). It never changes.
Why should that be true? Here's the picture I want you to keep in your head. Imagine a paper triangle. Tear off all three corners and slide them together so their points meet at a single spot. They fit perfectly along a straight line — and a straight line is \(180°\). The three angles, pooled together, make exactly a half-turn.
If you'd like the cleaner argument: draw a line through the top vertex that is parallel to the base. The two base angles reappear up at the top as alternate angles (the "Z" angles from the previous lesson), sitting on either side of the top angle. Together those three angles fill the straight line through the vertex — so they total \(180°\).
Drag any corner of the triangle below. Watch the three angles change wildly — but keep your eye on their sum.
In words The angles of a triangle always share out a half-turn. So if you know two of them, the third is never a mystery — just subtract the two you have from \(180°\).
- All three angles sum to \(180°\): \(50° + 70° + x = 180°\).
- Add the two you know: \(50° + 70° = 120°\).
- Subtract from \(180°\): \(x = 180° - 120° = 60°\).
The third angle is \(60°\). (With all three equal-ish here — \(50, 60, 70\) — you can already feel this is a fairly "ordinary" triangle.)
Naming triangles by their sides
Mathematicians like names that tell you something. There are exactly three families when you sort triangles by side lengths:
- Equilateral — all three sides equal. Because the sides match, the angles must too, so every angle is \(180° \div 3 = 60°\). Perfectly balanced.
- Isosceles — exactly two sides equal. The two angles opposite those equal sides are also equal. This "base angles are equal" fact is one you'll use constantly.
- Scalene — no two sides equal, and therefore no two angles equal. The most lopsided of the three.
Naming triangles by their angles
The same triangle can also be sorted by its biggest angle. There are three cases:
- Acute — all three angles are less than \(90°\). Nicely "pointy" all round.
- Right — one angle is exactly \(90°\). The square corner. (These are special enough to get their own lesson on the Pythagorean theorem later.)
- Obtuse — one angle is greater than \(90°\). It leans open. Note a triangle can have at most one angle that is right or obtuse — two would already use up \(180°\) on their own.
Try this in the widget above: drag a corner until one angle reads \(90°\), then push past it. Watch the label switch from acute to right to obtuse as the largest angle crosses each threshold.
Tip — two names, not a contradiction. Every triangle gets one side-name and one angle-name. "Right isosceles" or "acute scalene" are perfectly normal descriptions — they're just answering two different questions about the same shape.
The exterior angle shortcut
Extend one side of a triangle past a vertex and you create an exterior angle. There's a tidy rule for it: the exterior angle equals the sum of the two remote interior angles — the two angles it doesn't touch.
It falls straight out of the \(180°\) rule. Say the interior angle at that vertex is \(c\), and the other two are \(a\) and \(b\). The interior angle and its exterior angle sit on a straight line, so the exterior angle is \(180° - c\). But \(a + b + c = 180°\), which rearranges to \(a + b = 180° - c\). Those are the same thing — so the exterior angle equals \(a + b\). You can find it without ever computing \(c\) at all.
Congruence: when two triangles are truly identical
Now the powerful idea. Two triangles are congruent if they have the same shape and the same size — one is an exact copy of the other, even if it's been slid, turned, or flipped over. Place one on top of the other and every side and every angle would line up perfectly.
The clever part: you don't need to check all six measurements (three sides, three angles). A small set of matching parts is enough to lock in the whole triangle. These are the congruence tests:
- SSS — all three pairs of sides are equal. Three fixed side lengths can build only one triangle.
- SAS — two sides and the included angle (the one between them) are equal. The angle pins the two sides into one rigid shape.
- ASA — two angles and the side between them are equal. The two angles fix the directions; the shared side fixes the scale.
- AAS — two angles and a side not between them. This works too, because once you know two angles you know the third (they sum to \(180°\)), so it quietly becomes ASA.
- RHS — for right-angled triangles only: the Right angle, the Hypotenuse, and one other Side match. The right angle plus those two sides leave no freedom.
In words A congruence test is a minimum recipe. If a pair of triangles shares the right combination of parts, they must be identical — there's only one triangle that fits the recipe. Beware the trap: AAA is not a test. Three matching angles guarantee the same shape but say nothing about size — a small triangle and a giant one can have identical angles.
Congruent versus similar
That last warning points to a second, related idea. Two triangles are similar when they have the same shape but possibly different sizes — one is a scaled photocopy of the other. All matching angles are equal, and all matching sides are in the same ratio.
So the relationship is clean: congruent means same shape and same size (scale factor \(1\)); similar means same shape, any size. Every congruent pair is also similar, but not the other way round.
- List what matches: a side (\(6\) cm), a side (\(8\) cm), and the angle between them (\(40°\)).
- A side, the included angle, then another side — that's the S-A-S pattern.
- SAS is a valid congruence test, so the recipe is complete: the angle locks the two given sides into one fixed shape.
Yes — they are congruent by SAS. (If the \(40°\) had instead been at one end, not between the two sides, this would be the unreliable "SSA" arrangement, which can produce two different triangles — so always check the angle is the included one.)
Practice
Try each one yourself, then reveal the full solution.
1. A triangle has two angles measuring \(40°\) and \(75°\). What is the third angle?
The angles of any triangle sum to \(180°\), so subtract the two you know:
\[ 180° - 40° - 75° = 65°. \]The third angle is \(65°\). (All three are under \(90°\), so this is an acute triangle.)
2. Classify the triangle with sides \(5\), \(5\) and \(8\) — both by its sides and, as far as you can tell, by its angles.
Two of the sides are equal (\(5\) and \(5\)) and the third (\(8\)) is different. Exactly two equal sides means it is isosceles.
By angles: the longest side, \(8\), sits opposite the largest angle. Since \(8\) is noticeably longer than the two \(5\)s, that top angle is wide — in fact \(5^2 + 5^2 = 50\) is less than \(8^2 = 64\), which (looking ahead to Pythagoras) means the angle opposite the \(8\) is obtuse. So it is an obtuse isosceles triangle.
3. In two triangles, two sides and the angle between those two sides are equal. Which congruence test applies, and are the triangles congruent?
Two sides with the included angle between them gives the pattern side–angle–side, which is the SAS test.
SAS is one of the valid congruence tests: the angle fixes the directions of both sides and their shared corner, leaving only one possible triangle. So yes — the triangles are congruent (by SAS).