Angle Geometry
Angle geometry in Year 8 is the angle chase: a chain of short steps, each one an equation with its reason. This NSW Year 8 maths subtopic (NESA outcome MA4‑ANG‑C‑01) builds on facts from Year 7, such as angles on a straight line adding to \(180^\circ\), angles at a point adding to \(360^\circ\) and vertically opposite angles being equal.
You will find complementary and supplementary angles, unknown angles where lines cross, and angles written as expressions such as \(x + 2x + (x + 14) = 180\). You will also use the fact that a triangle's angles add to \(180^\circ\), the equal base angles of an isosceles triangle, the exterior angle of a triangle and angle bisectors, and find the angle between an ironing board's legs and the floor. It is part of Year 8 Geometry.
Theory
An angle chase is a chain of short steps. Each step is one equation with its reason in brackets, and the answer to one step is the given for the next: \(\angle KLM + 118 = 180\) (angles on a straight line), so \(\angle KLM = 62^\circ\). This Year 8 maths page shows how.
Complementary angles add to \(90^\circ\); supplementary angles add to \(180^\circ\). The complement of \(37^\circ\) is \(53^\circ\) and its supplement is \(143^\circ\).
Angles on a straight line add to \(180^\circ\). Angles in a right angle add to \(90^\circ\). Angles at a point (a revolution) add to \(360^\circ\); an unknown angle there is reflex when the others add to less than \(180^\circ\): with one angle of \(140^\circ\), the other way round is \(220^\circ\).
Vertically opposite angles are equal: where two straight lines cross, the angles facing each other match.
Triangle facts. The angle sum of a triangle is \(180^\circ\); the base angles of an isosceles triangle are equal; an exterior angle of a triangle equals the sum of the two interior opposite angles.
Naming. \(\angle ABC\) is the angle at \(B\), between the arms \(BA\) and \(BC\). A ray that bisects an angle cuts it into two equal halves; halving an odd number of degrees gives a decimal.
The chase. Pick the fact that links a known angle to an unknown one, write the equation and its reason on the same line, solve, and repeat.
Angles as expressions. Put the expressions into one equation and collect like terms: \(x + 3x + (x + 30) = 180\) gives \(5x + 30 = 180\), so \(x = 30\). “Find \(x\)” is not “find the angle”: substitute back.
The facts and the reasons to write:
| Fact | Reason to write |
|---|---|
| \(a + b = 180\) | angles on a straight line |
| \(a + b = 90\) | angles in a right angle |
| \(a + b + c = 360\) | angles at a point |
| \(a = b\) where two lines cross | vertically opposite angles |
| \(a + b + c = 180\) in a triangle | angle sum of a triangle |
| \(e = a + b\) | exterior angle of a triangle |
How to chase an angle
- Mark every angle you know on the diagram, including right angles and equal angles.
- Find a fact that links a known angle to an unknown one.
- Write the equation with its reason, and solve it.
- Repeat until you reach the angle asked for, then check that each sum works.
(i) its complement
(ii) its supplement
| \(90 - 37\) | \(=\) | \(53\) |
| \(180 - 37\) | \(=\) | \(143\) |
(i) \(53^\circ\)
(ii) \(143^\circ\)
The right-angle mark is worth \(90^\circ\). \(x + 142 + 90 = 360\) (angles at a point).
| \(x + 232\) | \(=\) | \(360\) |
| \(x\) | \(=\) | \(360 - 232 = 128\) |
\(x = 128\). Check: \(128 + 142 + 90 = 360\).
(i) \(x\)
(ii) \(z\)
(i) \(4x - 9 = 2x + 31\) (vertically opposite angles).
(ii) The angle is \(4 \times 20 - 9 = 71^\circ\), and \(z + 71 = 180\) (angles on a straight line).
| \(2x\) | \(=\) | \(40\) |
| \(x\) | \(=\) | \(20\) |
| \(z\) | \(=\) | \(180 - 71 = 109\) |
(i) \(x = 20\)
(ii) \(z = 109\). Check: \(2 \times 20 + 31 = 71\).
\(\angle ABD = 134 \div 2 = 67^\circ\) (\(BD\) bisects \(\angle ABC\)). Then \(\angle DAB + 67 + 79 = 180\) (angle sum of a triangle).
| \(\angle DAB\) | \(=\) | \(180 - 146 = 34^\circ\) |
\(\angle DAB = 34^\circ\). Check: \(34 + 67 + 79 = 180\).
(i) \(y\)
(ii) \(\angle ABC\)
(i) \(5y - 9 = (2y + 6) + (y + 15)\) (exterior angle of a triangle).
(ii) \(\angle CBD = 5 \times 15 - 9 = 66^\circ\), and \(\angle ABC + 66 = 180\) (angles on a straight line).
| \(5y - 9\) | \(=\) | \(3y + 21\) |
| \(2y\) | \(=\) | \(30\) |
| \(y\) | \(=\) | \(15\) |
| \(\angle ABC\) | \(=\) | \(180 - 66 = 114^\circ\) |
(i) \(y = 15\)
(ii) \(\angle ABC = 114^\circ\). Check: \(36 + 114 + 30 = 180\).
| \(x + 3x + (x + 30)\) | \(=\) | \(180\) |
| \(5x + 30\) | \(=\) | \(180\) |
| \(x\) | \(=\) | \(30\) |
The angles are \(30^\circ\), \(90^\circ\) and \(60^\circ\), so the largest is \(90^\circ\). Find \(x\) first, then substitute back.
Common pitfalls
Frequently asked questions
What are complementary angles?
Two angles that add to \(90^\circ\), such as \(30^\circ\) and \(60^\circ\).
What are supplementary angles?
Two angles that add to \(180^\circ\), such as \(110^\circ\) and \(70^\circ\).
What do angles at a point add to?
\(360^\circ\), a full revolution.
What are vertically opposite angles?
The pairs of angles facing each other where two straight lines cross. They are equal.
Why do I need to write reasons?
A reason shows which angle fact makes each step true, so the working can be checked.