Use the angle sum of a triangle and a quadrilateral, the exterior angle property, and the properties of special triangles and quadrilaterals

What this dot point is asking

SEAB wants you to use the angle sum of a triangle ($180^\circ$) and of a quadrilateral ($360^\circ$), the exterior angle property of a triangle, and the side and angle properties of special triangles and quadrilaterals. These facts, combined with reasons, let you find unknown angles in many shapes.

The answer

Angle sums

Two facts you will use constantly:

• The angles in a triangle add up to $180^\circ$.
• The angles in a quadrilateral add up to $360^\circ$.

So in a quadrilateral with three known angles of $90^\circ$, $100^\circ$ and $80^\circ$, the fourth is $360^\circ - 270^\circ = 90^\circ$.

Types of triangle

• An equilateral triangle has three equal sides and three $60^\circ$ angles.
• An isosceles triangle has two equal sides and two equal (base) angles.
• A right-angled triangle has one $90^\circ$ angle.
• A scalene triangle has all sides and angles different.

The equal-angle property of the isosceles triangle is especially common in exam questions.

The exterior angle property

If you extend one side of a triangle, the exterior angle formed equals the sum of the two interior angles not next to it:

This is often quicker than finding the interior angle first.

Special quadrilaterals

Each special quadrilateral has its own properties:

• Square: four equal sides, four right angles.
• Rectangle: opposite sides equal, four right angles.
• Parallelogram: opposite sides parallel and equal, opposite angles equal.
• Rhombus: four equal sides, opposite angles equal.
• Trapezium: one pair of parallel sides.

Knowing which properties belong to which shape tells you which angles or sides are equal.

Examples in context

Example 1. A roof truss. A symmetrical roof truss is an isosceles triangle, so its two base angles are equal. Knowing one base angle immediately gives the other and, with the angle sum, the apex angle at the top of the roof. The isosceles property does most of the work.

Example 2. A kite's angles. A kite is a quadrilateral, so its four angles sum to $360^\circ$. If three angles are known, the fourth follows by subtraction. Treating any four-sided figure as a quadrilateral lets you find a missing angle even when the shape is unusual.

Try this

• Cue. A triangle has angles $x$, $x$ and $40^\circ$. Then $2x + 40 = 180$, so $x = 70^\circ$.
• Cue. A quadrilateral has angles $90^\circ$, $90^\circ$, $120^\circ$ and $y$. Then $y = 360^\circ - 300^\circ = 60^\circ$.
• Cue. A triangle's exterior angle is $130^\circ$ and one opposite interior angle is $60^\circ$. The other is $130^\circ - 60^\circ = 70^\circ$.