Angle Sum of Triangles and Quadrilaterals
The angles of a triangle add to \(180^\circ\) and the angles of a quadrilateral add to \(360^\circ\), because a diagonal splits a quadrilateral into two triangles. This NSW Year 8 maths subtopic (NESA outcome MA4‑GEO‑C‑01) also uses the exterior angle of a triangle, which is the same size as the two opposite interior angles added together.
You will solve for angles given as expressions, including isosceles triangles and pronumerals on both sides, find corners made of two parts, and share an angle sum in a ratio such as \(2 : 3 : 4\). You will also work with non-convex quadrilaterals, decide whether a set of angles is possible, and find the corners of a farmer's paddock. It is part of Year 8 Geometry.
Theory
The angles of a triangle add to \(180^\circ\) and the angles of a quadrilateral add to \(360^\circ\). An exterior angle of a triangle equals the sum of the two interior opposite angles. Write the reason beside each line. This Year 8 maths page shows how.
Angle sum of a triangle: \(a + b + c = 180^\circ\).
Angle sum of a quadrilateral: \(a + b + c + d = 360^\circ\). A diagonal splits a quadrilateral into two triangles, so \(2 \times 180 = 360\). In a non-convex quadrilateral use the diagonal from the reflex corner, which lies inside.
Exterior angle of a triangle. Extend one side. The angle outside equals the sum of the two interior opposite angles: \(e = a + b\). It follows from \(e + c = 180\) (angles on a straight line) and \(a + b + c = 180\).
A corner can be made of parts. A diagonal splits a corner in two; work out each part in its own triangle, then add.
Equal sides, equal angles. In an isosceles triangle call both base angles \(x^\circ\): the equation becomes \(2x + \ldots = 180\).
Pronumerals on both sides: collect like terms, then subtract the smaller pronumeral term. Angles in a ratio: add the parts, divide the angle sum by that total, then multiply.
A reflex corner counts. A non-convex quadrilateral’s reflex angle is one of its four angles, and the sum is still \(360^\circ\).
Angle sums and reasons:
| Fact | Reason to write |
|---|---|
| \(a + b + c = 180\) | angle sum of a triangle |
| \(a + b + c + d = 360\) | angle sum of a quadrilateral |
| \(e = a + b\) | exterior angle of a triangle |
| base angles equal | base angles of an isosceles triangle |
How to find a missing angle
- Decide: triangle (\(180^\circ\)) or quadrilateral (\(360^\circ\)); change any outside angle to the inside one first.
- Write the equation with every angle, including right angles, and its reason.
- Collect like terms, solve, and check the angles add correctly.
| \(3a + 74 + 118 + 90\) | \(=\) | \(360\) |
| \(3a + 282\) | \(=\) | \(360\) |
| \(3a\) | \(=\) | \(78\) |
| \(a\) | \(=\) | \(26\) |
\(a = 26\) (angle sum of a quadrilateral). Check: \(78 + 282 = 360\).
| \(7x + 5\) | \(=\) | \(52 + (3x + 9)\) |
| \(7x + 5\) | \(=\) | \(3x + 61\) |
| \(4x\) | \(=\) | \(56\) |
| \(x\) | \(=\) | \(14\) |
\(x = 14\) (exterior angle of a triangle). Check: \(7 \times 14 + 5 = 103\) and \(52 + 51 = 103\).
(i) \(\angle BCD\)
(ii) \(x\)
| \(\angle BCD\) | \(=\) | \(180 - 68 = 112^\circ\) |
| \(x + 293\) | \(=\) | \(360\) |
| \(x\) | \(=\) | \(67\) |
(i) \(112^\circ\) (angles on a straight line)
(ii) \(x + 95 + 86 + 112 = 360\) (angle sum of a quadrilateral), so \(x = 67\).
(i) Find the smallest and the largest angle.
(ii) Is the quadrilateral convex?
| \(1 + 2 + 3 + 4\) | \(=\) | \(10\) |
| \(360 \div 10\) | \(=\) | \(36\) |
| \(4 \times 36\) | \(=\) | \(144\) |
(i) \(36^\circ\) and \(144^\circ\) (the angles are \(36^\circ\), \(72^\circ\), \(108^\circ\), \(144^\circ\))
(ii) Yes: no angle is reflex.
| \(2x + 48\) | \(=\) | \(180\) |
| \(2x\) | \(=\) | \(132\) |
| \(x\) | \(=\) | \(66\) |
\(\angle B = \angle C = 66^\circ\) (base angles of an isosceles triangle; angle sum of a triangle).
| \(x + 30 + 40 + 50\) | \(=\) | \(360\) |
| \(x\) | \(=\) | \(240\) |
\(x = 240\). The reflex angle is one of the four angles, so the sum is still \(360^\circ\).
Common pitfalls
Frequently asked questions
What do the angles of a triangle add to?
\(180^\circ\).
What do the angles of a quadrilateral add to?
\(360^\circ\), because a diagonal splits it into two triangles.
What is the exterior angle of a triangle?
The angle between one side and the extension of the next side. It equals the sum of the two interior opposite angles.
Can a quadrilateral have three obtuse angles?
Yes, for example \(100^\circ\), \(100^\circ\), \(100^\circ\) and \(60^\circ\).
How do I share an angle sum in a ratio?
Add the parts, divide the sum by that total, then multiply each part: \(1 : 2 : 3 : 4\) of \(360^\circ\) gives \(36^\circ\), \(72^\circ\), \(108^\circ\), \(144^\circ\).