Basic Geometry
Basic geometry uses angle facts to find unknown angles and to give a reason for every step: angles on a straight line and at a point, vertically opposite angles, angles on parallel lines, and the angle sums of triangles, quadrilaterals and polygons. In NSW Year 10 Advanced maths these Stage 4 facts are used in longer chains of reasoning (NESA outcome MA5‑GEO‑P‑01).
You will find unknown angles in diagrams with intersecting and parallel lines, isosceles triangles, parallelograms and trapeziums, with a reason for each step. You will use exterior angles of triangles and polygons, find the number of sides of a regular polygon, and solve equations for angles. Harder questions prove that a quadrilateral's angles add to \(360^\circ\) and that two lines are parallel. It is part of Year 10 Advanced Geometric Figures.
Theory
This Year 10 Advanced maths page revises the angle facts from earlier years and pushes them further: parallel lines, isosceles triangles, exterior angles, polygon angle sums and regular polygons, with problems that chain several facts together and short proofs. Polygon exterior angles come up again if you are heading for the Extension path.
When a transversal cuts two parallel lines, corresponding angles (F) are equal, alternate angles (Z) are equal, and co-interior angles (C) add to \(180^\circ\). The reverse is also true: if alternate angles are equal, or co-interior angles add to \(180^\circ\), the lines are parallel.
Angles on a straight line add to \(180^\circ\); angles at a point add to \(360^\circ\); vertically opposite angles are equal.
The angles of a triangle add to \(180^\circ\). An exterior angle equals the sum of the two interior opposite angles. In an isosceles triangle the angles opposite the equal sides are equal, and if two angles are equal the sides opposite them are equal.
A polygon with \(n\) sides has angle sum \((n - 2) \times 180^\circ\). The exterior angles of any convex polygon add to \(360^\circ\). In a regular polygon every interior angle is equal and every exterior angle is \(\dfrac{360^\circ}{n}\).
Every step of a geometry answer needs a reason, such as "alternate angles, \(AB \parallel CD\)".
| Fact | Rule | Reason to write |
|---|---|---|
| F shape | equal | corresponding angles, \(AB \parallel CD\) |
| Z shape | equal | alternate angles, \(AB \parallel CD\) |
| C shape | add to \(180^\circ\) | co-interior angles, \(AB \parallel CD\) |
| triangle | add to \(180^\circ\) | angle sum of a triangle |
| exterior angle | \(=\) sum of interior opposite angles | exterior angle of a triangle |
| isosceles | base angles equal | base angles of isosceles triangle |
| quadrilateral | add to \(360^\circ\) | angle sum of a quadrilateral |
| parallelogram | opposite angles equal, co-interior add to \(180^\circ\) | opposite angles of a parallelogram |
How to find an angle with reasons
- Mark the diagram: equal sides, parallel lines and every angle you know.
- Look for a link between a known angle and the one you want: a straight line, a triangle, a pair of parallel lines, equal sides.
- Write one fact per line with its reason, naming the parallel lines or the triangle.
- Algebra: if the angles are expressions, write the equation the fact gives (equal, or adding to \(180^\circ\) or \(360^\circ\)), solve, then substitute back.
Harder situations
- A bend between parallel lines: draw a third parallel line through the bend (Example 1).
- Two transversals with two unknowns: each angle fact gives an equation; solve the pair simultaneously.
- Showing lines are parallel: find a pair of equal alternate or corresponding angles, or co-interior angles adding to \(180^\circ\), and state the conclusion.
- Showing a triangle is isosceles: find two equal angles; the sides opposite them are equal.
- Proofs of angle sums: split the shape with a diagonal or construct a parallel line, then add the known sums.
- Regular polygons: find the exterior angle first, often from interior \(+\) exterior \(= 180^\circ\), then \(n = 360^\circ \div e\).
No single pair of angles links the two lines. Draw \(CF\) through \(C\) parallel to \(AB\) and \(ED\), which splits \(\angle BCD\) in two.
\(\angle BCF = 48^\circ\) (alternate angles, \(AB \parallel CF\))
\(\angle FCD = 180^\circ - 130^\circ = 50^\circ\) (co-interior angles, \(CF \parallel ED\))
\(\angle BCD = 48^\circ + 50^\circ = \mathbf{98^\circ}\)
Check which angle pair you have: \(\angle CDE\) and \(\angle FCD\) sit between the parallel lines on the same side of \(CD\), so they are co-interior, not alternate.
Work from the given angle into the triangle, one fact at a time.
\(\angle ACB = 35^\circ\) (alternate angles, \(AE \parallel BC\))
\(\angle BAC = 35^\circ\) (base angles of isosceles \(\triangle ABC\), \(AB = BC\))
\(\angle ABC = 180^\circ - 35^\circ - 35^\circ = \mathbf{110^\circ}\) (angle sum of \(\triangle ABC\))
The equal angles are opposite the equal sides: \(\angle BAC\) faces \(BC\) and \(\angle ACB\) faces \(AB\).
(i) Each interior angle of a regular polygon is \(8\) times each exterior angle. How many sides does it have?
(ii) A pentagon has angles \(3x^\circ\), \(3x^\circ\), \((2x + 10)^\circ\), \((2x + 10)^\circ\) and \(120^\circ\). Find \(x\) and the largest angle.
(i) An interior angle and its exterior angle lie on a straight line. Let the exterior angle be \(e\).
| \(8e + e\) | \(=\) | \(180^\circ\) |
| \(e\) | \(=\) | \(20^\circ\) |
| \(n\) | \(=\) | \(\dfrac{360^\circ}{20^\circ} = \mathbf{18} \text{ sides}\) |
The exterior angles of any convex polygon add to \(360^\circ\).
(ii) The angle sum of a pentagon is \((5 - 2) \times 180^\circ = 540^\circ\).
| \(3x + 3x + (2x + 10) + (2x + 10) + 120\) | \(=\) | \(540\) |
| \(10x + 140\) | \(=\) | \(540\) |
| \(x\) | \(=\) | \(\mathbf{40}\) |
The angles are \(120^\circ, 120^\circ, 90^\circ, 90^\circ, 120^\circ\). The largest is \(\mathbf{120^\circ}\). Check: they add to \(540^\circ\).
Construct \(CE\) through \(C\) parallel to \(BA\). Let \(\angle BAC = a\) and \(\angle ABC = b\).
\(\angle ACE = a\) (alternate angles, \(BA \parallel CE\))
\(\angle ECD = b\) (corresponding angles, \(BA \parallel CE\))
\(\angle ACD = \angle ACE + \angle ECD = a + b\)
So \(\angle ACD = \angle BAC + \angle ABC\), as required. Every line of a proof states a fact and its reason; a construction line is allowed if you say how it was drawn.
Common pitfalls
Frequently asked questions
What is the difference between alternate and co-interior angles?
Both sit between two parallel lines. Alternate angles are on opposite sides of the transversal, in a Z shape, and are equal. Co-interior angles are on the same side, in a C shape, and add to \(180^\circ\).
How do you find the angle sum of a polygon?
Multiply \(180^\circ\) by two less than the number of sides. A hexagon has \(6\) sides, so its angles add to \(4 \times 180^\circ = 720^\circ\).
How do you find the number of sides of a regular polygon?
Find one exterior angle, then divide \(360^\circ\) by it. If each interior angle is \(140^\circ\), each exterior angle is \(40^\circ\) and there are \(9\) sides.
What is the exterior angle of a triangle equal to?
The sum of the two interior angles that are not next to it, called the interior opposite angles.
How do you prove two lines are parallel?
Show that a pair of alternate angles or corresponding angles are equal, or that a pair of co-interior angles add to \(180^\circ\), then state that the lines are parallel.