Triangles: Properties, Types & Key Theorems: quick revision, 15 bites

The 180° Rule That Defines Every Triangle

Three angles, one unbreakable sum.

The **angle sum property** is the single most important fact about triangles — it holds for every triangle, no matter its shape or size.

Takeaway. Knowing any two interior angles instantly gives you the third — this is the fastest tool in triangle problems.

What Exactly Is a Triangle?

Three sides, three angles, one closed shape.

Takeaway. A triangle is defined by three non-collinear points — collinear points cannot form a triangle.

Classifying Triangles by Sides

Equal sides change everything.

A third type — **scalene** — has all three sides (and all three angles) different.

Takeaway. Side equality directly forces angle equality — equilateral is a special isosceles, but not vice versa.

Classifying Triangles by Angles

One angle decides the whole type.

Takeaway. The largest angle alone determines whether a triangle is acute, right, or obtuse.

The Exterior Angle Theorem

Outside angle equals two inside angles.

This follows directly from the 180° rule: the exterior angle and its adjacent interior angle are supplementary, and the two remote interior angles also sum to the same value.

Takeaway. An exterior angle is always greater than either of its two remote interior angles.

Triangle Inequality: The Side Rule

Two sides must beat the third.

Takeaway. Use the triangle inequality to test whether three given lengths can actually form a triangle.

Congruence Is Not Similarity

Same shape ≠ same size.

Takeaway. Congruence implies similarity, but similarity does NOT imply congruence.

The Four Congruence Criteria

Four tests to prove identical triangles.

Takeaway. AAA and SSA are NOT congruence criteria — they don't guarantee identical triangles.

Pythagoras: The Right Triangle Rule

Squares of legs add to hypotenuse.

The hypotenuse is always the **longest side** and lies opposite the 90° angle. This theorem works only for right triangles.

Takeaway. If a² + b² = c² holds, the triangle is right-angled — this is also a test for right triangles.

Finding the Missing Angle

Two angles in, third one out.

Takeaway. Always subtract the known angles from 180° — never from 360°.

Triangles Hold the World Together

Why bridges love triangles.

Takeaway. The triangle is the only polygon that cannot be deformed without changing side lengths — that's why engineers rely on it.

Remember the Angle Sum

Three-sixty split in half.

A full turn is **360°**. A triangle is half of a quadrilateral's angle sum (360°), so its angles add to **180°**. Picture folding a square diagonally — two triangles, each with 180°.

Takeaway. Link 180° to half of 360° — it makes the number impossible to forget.

Exam Shortcuts for Triangle Problems

Score faster with these moves.

Takeaway. Identify the triangle type first — it narrows down which theorem or criterion to apply.

The Triangle That Isn't

Three lengths, no triangle.

If you try to build a triangle with sides 2 cm, 3 cm, and 6 cm, it's impossible — because 2 + 3 = 5 < 6. The triangle inequality fails, so the three segments can never close into a triangle.

Takeaway. Always test the triangle inequality before assuming three lengths form a triangle.

Similar Triangles in Real Measurement

Measure a tree without climbing it.

Takeaway. Similar triangles turn indirect measurement into simple ratio problems.