# Triangles

> Name triangles by their sides and angles, use the 180 degree rule, isosceles triangles and the exterior angle.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/triangles/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The paper corners
Ms Rossi hands every student a paper triangle. "Tear off the three corners," she says. "Now put the three points together so they touch."

Ruby fits her three torn corners side by side. They make a perfectly straight line! Everyone's triangle is a different shape, but everyone gets a straight line.

A straight line is $180^\circ$. So the three angles of **any** triangle add up to $180^\circ$. That one fact solves a huge number of test questions.

**In short:** The three angles in any triangle add to $180^\circ$. An equilateral triangle has 3 equal sides and three $60^\circ$ angles, an isosceles triangle has 2 equal sides and 2 equal angles, and a scalene triangle has no equal sides.

## Before you start

- A straight line is $180^\circ$ and a right angle is $90^\circ$ (see Types of Angles).
- Acute, right and obtuse angles.
- Equal sides in a diagram are shown with matching little dashes called **tick marks**.

## Angle sum of a triangle

### Key idea: Always $180^\circ$
The three angles inside any triangle add up to $180^\circ$.

To find a missing angle: $180^\circ - (\text{the other two})$.

### Worked example: Missing angle
A triangle has angles $81^\circ$ and $76^\circ$. The third is $180 - 81 - 76 = 23^\circ$.

## Naming triangles by their sides

*Diagram: Three triangles: equilateral with three equal sides, isosceles with two equal sides, and scalene with no equal sides*
*Tick marks show which sides are equal.*

| Name | Sides | Angles |
|---|---|---|
| **Equilateral** | all 3 equal | all 3 equal, each $60^\circ$ |
| **Isosceles** | 2 equal | the 2 angles opposite the equal sides are equal |
| **Scalene** | none equal | all 3 different |

### Key idea: Isosceles: equal sides, equal base angles
In an isosceles triangle, the two angles at the ends of the third side (the **base angles**) are equal.

If the top angle is $36^\circ$, the two base angles share $180 - 36 = 144$, so each is $72^\circ$.

If one base angle is $40^\circ$, the other is also $40^\circ$, and the top is $180 - 80 = 100^\circ$.

## Naming triangles by their angles

- **Acute-angled:** all three angles are acute.
- **Right-angled:** one angle is $90^\circ$. The other two add to $90^\circ$. The longest side, opposite the right angle, is called the **hypotenuse**.
- **Obtuse-angled:** one angle is obtuse.

A triangle can only have **one** right angle or **one** obtuse angle, because the other two must still fit into $180^\circ$.

### Worked example: Right-angled
A right-angled triangle has an angle of $37^\circ$. The third angle is $90 - 37 = 53^\circ$.

A right-angled isosceles triangle has two equal angles sharing $90^\circ$, so each is $45^\circ$.

## The exterior angle

If you stretch one side of a triangle past a corner, you make an **exterior angle** outside the triangle.

*Diagram: Triangle with interior angles a and b, and an exterior angle at the third corner equal to a plus b*
*The exterior angle equals the two inside angles at the other corners added together.*

### Key idea: Exterior angle rule
An exterior angle of a triangle equals the **sum of the two interior angles opposite it**.

Why? The exterior angle and the third inside angle make a straight line of $180^\circ$, and so do all three inside angles. Take the third angle away from both and you are left with exterior $= a + b$.

### Worked example: Using the shortcut
Inside angles of $40^\circ$ and $60^\circ$. The exterior angle at the third corner is $40 + 60 = 100^\circ$.

### Challenge: Selective level: isosceles plus exterior angles
An isosceles triangle has an exterior angle of $116^\circ$ at one base corner. The base angle inside is $180 - 116 = 64^\circ$. The other base angle is also $64^\circ$. The top angle is $180 - 128 = 52^\circ$.

An isosceles triangle is cut in half by a line from the top to the middle of the base. The line meets the base at a right angle. If a base angle is $46^\circ$, each half of the top angle is $90 - 46 = 44^\circ$.

The 3-4-5 triangle: a triangle with sides 3 cm, 4 cm and 5 cm always has a right angle between the 3 cm and 4 cm sides. Builders use it to make square corners.

**Mind map**

```text
Triangles
  Angle sum
    Always 180°
  By sides
    Equilateral: all 60°
    Isosceles: base angles equal
    Scalene: all different
  By angles
    Acute
    Right: other two add to 90°
    Obtuse
  Exterior angle
    Sum of the two opposite inside angles
```

## Quick summary

- The angles in a triangle add to $180^\circ$.
- Equilateral: 3 equal sides and three $60^\circ$ angles.
- Isosceles: 2 equal sides and 2 equal base angles.
- Scalene: no equal sides or angles.
- In a right-angled triangle, the other two angles add to $90^\circ$. The longest side is the hypotenuse.
- An exterior angle equals the sum of the two opposite interior angles.

## Quiz

Try these questions from the chapter quiz.

1. A triangle has angles $81^\circ$ and $76^\circ$. What is the third angle?
   - A. $13^\circ$
   - B. $23^\circ$
   - C. $33^\circ$
   - D. $157^\circ$
   - Answer: B. $23^\circ$. $180 - 81 - 76 = 23$.

2. What is each angle in an equilateral triangle?
   - A. $45^\circ$
   - B. $60^\circ$
   - C. $90^\circ$
   - D. $180^\circ$
   - Answer: B. $60^\circ$. Three equal angles share $180^\circ$: $180 \div 3 = 60$.

3. A right-angled triangle has an angle of $37^\circ$. What is the third angle?
   - A. $63^\circ$
   - B. $143^\circ$
   - C. $53^\circ$
   - D. $43^\circ$
   - Answer: C. $53^\circ$. The two smaller angles add to $90^\circ$: $90 - 37 = 53$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/triangles/#quiz