Triangle Gemoetry
Triangle geometry uses a few facts to find unknown angles: the angles of a triangle add to \(180^\circ\), an exterior angle equals the two opposite interior angles added together, and an isosceles triangle has equal base angles. This NSW Year 9 Core maths subtopic revises Stage 4 work under NESA outcome MA4‑GEO‑C‑01.
You will find a missing angle, the angles of equilateral and isosceles triangles, and classify a triangle by its sides and its angles at the same time, such as scalene and right-angled. You will also form equations from angles written as \(x\), \(2x\) and \(3x\) or in a ratio such as \(2 : 3 : 5\), use the exterior angle fact forwards and backwards, and reach the angle you want through a straight angle, vertically opposite angles or parallel lines first. It is part of Year 9 Core Properties of Geometric Figures.
Theory
The three angles of any triangle add to 180°. This Year 9 maths page revises the earlier triangle facts and uses them a step further: isosceles and equilateral triangles, the exterior angle theorem, angles given with pronumerals or as a ratio, and classifying a triangle by its angles and its sides.
The angle sum of a triangle is \(180^\circ\). The three interior angles always add to \(180^\circ\).
By its angles, a triangle is acute-angled (all angles less than \(90^\circ\)), right-angled (one angle of \(90^\circ\)) or obtuse-angled (one angle more than \(90^\circ\)).
By its sides, a triangle is equilateral (three equal sides), isosceles (two equal sides) or scalene (no equal sides). Equal sides are marked with matching ticks.
In an isosceles triangle, the two angles opposite the equal sides are equal. They are the base angles. The third angle is the apex. In an equilateral triangle every angle is \(60^\circ\).
Extend one side and the angle outside is an exterior angle. It equals the sum of the two interior angles furthest from it.
The two main rules:
| By its angles | By its sides | ||
|---|---|---|---|
| acute-angled | all three less than \(90^\circ\) | equilateral | all three sides equal |
| right-angled | one angle of \(90^\circ\) | isosceles | two sides equal |
| obtuse-angled | one angle more than \(90^\circ\) | scalene | no sides equal |
How to find an unknown angle in a triangle
- Mark every angle you know on the diagram, including equal angles from equal sides.
- Bring in angles from outside the triangle using straight angles, vertically opposite angles or parallel lines.
- Use the angle sum, or the exterior angle theorem, to write an equation.
- Solve and check that the three angles add to \(180^\circ\).
(i) Two angles of a triangle are \(43^\circ\) and \(71^\circ\). Find the third angle.
(ii) The angles of a triangle are \(x\), \(x + 20^\circ\) and \(2x\). Find \(x\) and each angle.
(i) Add the two known angles and take them from \(180^\circ\).
| \(\text{third angle}\) | \(=\) | \(180^\circ - (43^\circ + 71^\circ)\) |
| \(\) | \(=\) | \(\mathbf{66^\circ}\) |
(ii) Add the three angles and set the total equal to \(180^\circ\).
| \(x + (x + 20) + 2x\) | \(=\) | \(180\) |
| \(4x + 20\) | \(=\) | \(180\) |
| \(4x\) | \(=\) | \(160\) |
| \(x\) | \(=\) | \(\mathbf{40}\) |
The angles are \(40^\circ\), \(60^\circ\) and \(80^\circ\). Check: \(40 + 60 + 80 = 180\) \(\checkmark\)
(i) An isosceles triangle has an apex angle of \(34^\circ\). Find each base angle.
(ii) Another isosceles triangle has base angles of \(38^\circ\). Find its apex angle.
(i) The base angles are equal. Call each one \(x\).
| \(34 + x + x\) | \(=\) | \(180\) |
| \(2x\) | \(=\) | \(146\) |
| \(x\) | \(=\) | \(\mathbf{73^\circ}\) |
(ii) Take both base angles from \(180^\circ\).
| \(\text{apex}\) | \(=\) | \(180^\circ - 2 \times 38^\circ\) |
| \(\) | \(=\) | \(\mathbf{104^\circ}\) |
(i) \(x\), the size of \(\angle ABC\)
(ii) \(\angle ACB\).
(i) The exterior angle equals the sum of the two interior opposite angles.
| \(52^\circ + x\) | \(=\) | \(115^\circ\) |
| \(x\) | \(=\) | \(\mathbf{63^\circ}\) |
(ii) \(\angle ACB\) and \(\angle BCD\) form a straight angle, so they add to \(180^\circ\).
| \(\angle ACB\) | \(=\) | \(180^\circ - 115^\circ = \mathbf{65^\circ}\) |
Check: \(52 + 63 + 65 = 180\) \(\checkmark\)
Count the parts, then share \(180^\circ\) between them.
| \(\text{parts}\) | \(=\) | \(1 + 3 + 5 = 9\) |
| \(\text{one part}\) | \(=\) | \(180^\circ \div 9 = 20^\circ\) |
| \(\text{angles}\) | \(=\) | \(20^\circ, \ 60^\circ, \ 100^\circ\) |
One angle is more than \(90^\circ\), so it is obtuse-angled. No two angles are equal, so no two sides are equal.
It is an obtuse-angled scalene triangle.
Common pitfalls
Frequently asked questions
What do the angles in a triangle add up to?
The three angles inside any triangle add up to \(180^\circ\).
How do you find the base angles of an isosceles triangle?
Take the apex angle away from \(180^\circ\), then halve the result. With an apex of \(34^\circ\), each base angle is \(146 \div 2 = 73^\circ\).
What is the exterior angle theorem?
An exterior angle of a triangle equals the sum of the two interior angles that are not next to it.
Can a triangle be both right-angled and isosceles?
Yes. A right-angled isosceles triangle has angles of \(90^\circ\), \(45^\circ\) and \(45^\circ\), with the two shorter sides equal.
How do you find angles given as a ratio?
Add the parts of the ratio, divide \(180^\circ\) by that total to get one part, then multiply each part of the ratio by it.