Angles

In the diagram the $68\degree$ angle on the top line has an alternate angle of $68\degree$ inside the triangle. With a known angle of $48\degree$ at the other corner, the triangle rule finishes the job: $180\degree - 68\degree - 48\degree = 64\degree$. Two rules, one tidy answer — and a reason written at each step.

• Line = 180°, point = 360°, vertically opposite angles equal.
• Parallel lines give equal F and Z angles, and C angles adding to 180°.
• Chain rules: find one angle, then step across to the next.
• Write a reason beside every angle you record.

A pentagon ($n = 5$) gives $540\degree$; a hexagon ($n = 6$) gives $720\degree$; an octagon ($n = 8$) gives $1080\degree$.

A regular polygon has all sides and all angles equal, so one interior angle is the total shared out: $\text{one interior angle} = \frac{(n - 2) \times 180\degree}{n}.$

Each angle of a regular hexagon is $720\degree \div 6 = 120\degree$. For an irregular polygon you still know the total $S$ — simply subtract the angles you are given to find a missing one.

• A polygon with n sides splits into (n − 2) triangles.
• Interior angle sum S = (n − 2) × 180°.
• Regular polygon: one interior angle = S ÷ n.
• Irregular polygon: subtract the known angles from S.

A regular octagon has $360\degree \div 8 = 45\degree$ at each corner. The interior and exterior angle at a corner sit on a straight line, so $\text{interior} = 180\degree - \text{exterior}$.

The real Stage 9 power is working backwards. If you know a regular polygon has an exterior angle of $24\degree$, then $n = 360\degree \div 24\degree = 15$ sides. If you are given an interior angle of $150\degree$, first find the exterior angle, $180\degree - 150\degree = 30\degree$, then $n = 360\degree \div 30\degree = 12$ sides.

• Exterior angles of any polygon add up to 360°.
• Regular polygon: one exterior angle = 360° ÷ n.
• Interior + exterior at a corner = 180°.
• Find n from an angle: n = 360° ÷ exterior angle.

To find the hypotenuse, add the squares of the two shorter sides and take the square root: $c = \sqrt{a^2 + b^2}.$

To find a shorter side, rearrange — subtract instead of add: $a = \sqrt{c^2 - b^2}.$

For a triangle with shorter sides $6\text{ cm}$ and $8\text{ cm}$: $c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\text{ cm}$.

A quick check: the hypotenuse must be the longest side. If your answer for $c$ comes out shorter than one of the other sides, something has gone wrong.

• Pythagoras' theorem works for every right-angled triangle.
• The hypotenuse is the longest side, opposite the right angle.
• Find the hypotenuse: c = √(a² + b²).
• Find a shorter side: subtract — a = √(c² − b²).

A $5\text{ m}$ ladder rests against a wall with its foot $1.5\text{ m}$ from the base. The ladder is the hypotenuse, and the height $h$ is a shorter side, so subtract: $h = \sqrt{5^2 - 1.5^2} = \sqrt{25 - 2.25} = \sqrt{22.75} \approx 4.77\text{ m}.$

The diagonal of a rectangle works the same way: the two sides are $a$ and $b$, and the diagonal is the hypotenuse $c$.

• Sketch the problem and mark the right angle clearly.
• Label the hypotenuse and the two shorter sides.
• Finding the hypotenuse → add; finding a shorter side → subtract.
• Take the square root and round to a sensible accuracy.

Pythagoras' theorem says $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

$\text{exterior} = 180 - 156 = 24$

The diagonal of a rectangle is the hypotenuse of a right-angled triangle, with the two sides as the shorter sides.