A friendly flat-vector scene of three triangles, one with equal sides and one with a right angle, in IllumiTutor navy and amber on an off-white background.

Triangles

Special triangle properties, the angle sum of 180 degrees, and orientation-independent base and vertex angles in isosceles triangles.

⏱ 9 min · 🎯 4 things to master

A triangle does not need to point upwards to keep its rules. Turn it sideways, choose a different base, or make one corner a right angle: its three interior angles still add to 180 degrees.

Parents: ask your child to name the equal sides or the chosen base before opening each angle solution.

By the end you will be able to recognise special triangles, use the 180-degree angle sum, name base and vertex angles correctly in a rotated isosceles triangle, and find unknown angles from the lines already shown.

Three special triangles

An has two equal sides and two equal angles. An has three equal sides and three equal angles. A has one right angle.

An equilateral triangle has three equal angles sharing 180 degrees, so each angle is 60 degrees. A right-angled triangle already gives one angle of 90 degrees; the other two still share the remaining 90 degrees.

🤔 Predict first: What is the size of every angle in an equilateral triangle?

Every triangle has an angle sum of 180 degrees

The three interior angles of any triangle add to 180 degrees. If two are known, subtract both from 180 degrees to find the third. No extra line needs to be drawn.

🤔 Predict first: A triangle has angles of 50 degrees, 60 degrees and x. What is x?

Explore the angles in a triangle

Predict first: What is the total of the three interior angles of a triangle?

Base angles and vertex angle in an isosceles triangle

In an isosceles triangle, the two equal sides meet at one corner. That corner contains the . The other two corners are the , because they sit at the endpoints of the base opposite the equal sides.

The base angles are defined by the chosen base and equal sides, not by whichever corners happen to look like the bottom of a page.

🤔 Predict first: In the shown isosceles triangle, which label names the vertex angle?

For the rotated example, the equal sides meet at V and the given vertex angle is 120 degrees. Therefore the two equal base angles at P and Q share 60 degrees, so each is 30 degrees.

Show the rotated isosceles solution

Vertex angle given

Two base angles=180∘−120∘=60∘\text{Two base angles}=180^\circ-120^\circ=60^\circ
Each base angle=60∘÷2=30∘\text{Each base angle}=60^\circ\div2=30^\circ

If a vertex angle is given, subtract it from 180 degrees and halve the remainder. If a base angle is given instead, subtract twice that base angle from 180 degrees to find the vertex angle. The subtract-then-halve method is only for finding the two equal base angles from a known vertex angle.

🤔 Predict first: An isosceles triangle has a vertex angle of 80 degrees. What is each base angle?

Use the properties of equilateral and right-angled triangles

For an equilateral triangle, all three angles are 60 degrees. For a right-angled triangle, one angle is 90 degrees, so the other two add to 90 degrees. These facts combine with the 180-degree triangle sum.

🤔 Predict first: A right-angled triangle has one other angle of 25 degrees. What is its third angle?

Find unknown angles from the lines already shown

Read the labels first. Use the triangle sum for angles inside a triangle. If the diagram also shows a straight line, angles on that straight line add to 180 degrees. Use a point-angle or vertically-opposite fact only when those rays are actually drawn.

For example, a triangle with angles 35 degrees and 65 degrees has a third angle of 80 degrees. If that 80-degree angle is shown beside another angle on a straight line, the neighbour is 100 degrees.

🤔 Predict first: A triangle has angles 45 degrees and 55 degrees. What is its third angle?

Show the worked solution

One triangle step

Unknown angle=180∘−45∘−55∘\text{Unknown angle}=180^\circ-45^\circ-55^\circ
=80∘=80^\circ

Three graduated practices

Keep every given visible while you choose the rule. Open each solution only after writing the angle fact you intend to use.

Practice 1 — the angle sum. A triangle has angles 55 degrees and 65 degrees. Find its third angle.

Reveal solution: Practice 1

Triangle angle sum

Third angle=180∘−55∘−65∘=60∘\text{Third angle}=180^\circ-55^\circ-65^\circ=60^\circ

The third angle is 60 degrees.

Practice 2 — a known vertex angle. An isosceles triangle has equal sides meeting at its vertex angle of 80 degrees. Find each base angle.

Reveal solution: Practice 2

Equal base angles

Two base angles=180∘−80∘=100∘\text{Two base angles}=180^\circ-80^\circ=100^\circ
Each base angle=100∘÷2=50∘\text{Each base angle}=100^\circ\div2=50^\circ

Each base angle is 50 degrees.

Practice 3 — the base angle is given. A rotated isosceles triangle has one base angle of 35 degrees. Find its vertex angle.

Reveal solution: Practice 3

Base angle given

Two base angles=2×35∘=70∘\text{Two base angles}=2\times35^\circ=70^\circ
Vertex angle=180∘−70∘=110∘\text{Vertex angle}=180^\circ-70^\circ=110^\circ

The vertex angle is 110 degrees. The triangle could be turned any way; the equal-angle rule still follows the equal sides.

Watch out — easily mixed up

Quick recap

🎯 Mastery check

Answer all 8 — your progress is saved on this device.

  1. What is the size of each angle in an equilateral triangle?

  2. A right-angled triangle has another angle of 25 degrees. What is its third angle?

  3. A triangle has angles 50 degrees and 60 degrees. What is its third angle?

  4. An isosceles triangle has vertex angle 80 degrees. What is each base angle?

  5. An isosceles triangle has each base angle equal to 65 degrees. What is its vertex angle?

  6. In rotated isosceles triangle VPQ, VP and VQ are equal and the angle at V is 120 degrees. What is the angle at P?

  7. A triangle has angles 45 degrees and 55 degrees. What is its third angle?

  8. Which statement correctly defines the vertex angle of an isosceles triangle?