Types of triangles

Before we measure or compute, let's name. A triangle can be classified in two ways: by sides and by angles.

Idea

By sides:

• Scalene , all three sides have different lengths.
• Isosceles , exactly two sides equal.
• Equilateral , all three sides equal.

By angles:

• Acute , all three angles are less than $90^\circ$.
• Right , one angle is exactly $90^\circ$.
• Obtuse , one angle is greater than $90^\circ$.

These classifications combine. So you can have a "right isosceles" triangle (two equal sides and a $90^\circ$ angle), an "obtuse scalene", an "acute equilateral", and so on. Note that an equilateral triangle is always acute (each angle is $60^\circ$).

A few facts that follow from the angle sum (next subtopic):

• A right triangle's other two angles sum to $90^\circ$. They are complementary.
• An obtuse triangle has only one obtuse angle (you cannot have two angles each greater than $90^\circ$, because then their sum would exceed $180^\circ$).
• An equilateral triangle has all angles equal to $60^\circ$.

In an isosceles triangle, the two angles opposite the equal sides are equal. This is the isosceles triangle property , a beautiful symmetry fact.

Worked examples

Example 1. Classify by sides: triangle with sides $5, 5, 5$.

All three equal , equilateral.

Example 2. Classify by angles: triangle with angles $50^\circ, 60^\circ, 70^\circ$.

All less than $90^\circ$ , acute.

Example 3. Classify (both ways): triangle with sides $3, 4, 5$.

All different , scalene. The angles? It is a famous right triangle ($3^2 + 4^2 = 5^2$). So right scalene.

Example 4. A triangle has two sides of length $7$ cm. What is the special name and what can be said about its angles?

Two sides equal , isosceles. The angles opposite those two equal sides are equal.

Try it yourself

1. Classify a triangle with sides $6, 8, 10$ (both by sides and by angles).
2. Classify a triangle with angles $30^\circ, 60^\circ, 90^\circ$.
3. Can a triangle be both isosceles and right? Give an example.
4. Can a triangle have two right angles? Why or why not?
5. Classify a triangle with sides $7, 7, 5$.
6. A triangle has angles $40^\circ, 70^\circ, 70^\circ$. Classify by sides.
7. Can an equilateral triangle be obtuse?
8. Sketch one example of each: scalene-acute, scalene-right, scalene-obtuse, isosceles-acute, isosceles-right.

Activity

Cut out paper triangles of three types. Measure each side and each angle. Verify the classifications above.