Angle Sum of Triangles and Quadrilaterals
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 \(36^\circ\), \(72^\circ\), \(108^\circ\), \(144^\circ\).