Angle facts, parallel lines and polygons
| English | Chinese | Pinyin |
|---|---|---|
| regular | 正的 | zhèng de |
| vertically opposite | 对顶角 | duì dǐng jiǎo |
| transversal | 截线 | jié xiàn |
| corresponding | 同位角 | tóng wèi jiǎo |
| alternate | 内错角 | nèi cuò jiǎo |
| interior angles | 内角 | nèi jiǎo |
| polygon | 多边形 | duō biān xíng |
| co-interior | 同旁内角 | tóng páng nèi jiǎo |
| exterior angles | 外角 | wài jiǎo |
Why do honeycombs have six sides?
- A regular 正的 hexagon tiles perfectly with no gaps — and uses the least wax per cell.
- Bees discovered this millions of years before mathematicians proved it. The angle facts behind it are what you'll learn here.
Parallel line and polygon lab
Pick the angle rule that unlocks each diagram.
Angle facts at a point and on a line
- Angles at a point add up to $360^{\circ}$.
- Angles on a straight line add up to $180^{\circ}$.
- Vertically opposite 对顶角 angles (where two lines cross) are equal.
Three angles on a line: $x + 50^{\circ} + 70^{\circ} = 180^{\circ} \Rightarrow x = 60^{\circ}$.
Give reasons in the exam. Writing "$60^{\circ}$" alone earns nothing. You must write "$60^{\circ}$, angles on a straight line add to $180^{\circ}$."
A honeycomb tessellates the plane with regular hexagons
Three angles on a straight line are x, 50° and 70°. Find x (degrees).
x + 50 + 70 = 180, so x = 60°.
Vertically opposite angles (where two lines cross) are always equal.
When two lines cross, the angles opposite each other are equal.
Angles in parallel lines
- When a transversal 截线 crosses two parallel lines:
- Corresponding 同位角 angles (F-shape) are equal.
- Alternate 内错角 angles (Z-shape) are equal.
- Co-interior angles 内角 (C-shape) add up to $180^{\circ}$.
Angles on a straight line add up to 180 degrees
A co-interior angle to 110° (between parallel lines) is how many degrees?
Co-interior angles add to 180°: 180 − 110 = 70°.
Polygons 多边形
- For a polygon with $n$ sides:
- A regular polygon has all sides and angles equal.
- Regular hexagon: exterior $= \dfrac{360}{6} = 60^{\circ}$, interior $= 180 - 60 = 120^{\circ}$.
Crossing two parallel lines: corresponding (F) and alternate (Z) angles are equal; co-interior 同旁内角 (C) angles add to $180^\circ$
Each interior angle of a regular hexagon is how many degrees?
Exterior = 360/6 = 60°, so interior = 180 − 60 = 120°.
The interior angles of a pentagon (5 sides) add up to how many degrees?
(n − 2) × 180 = (5 − 2) × 180 = 540°.
The sum of exterior angles 外角
- The exterior angles of any polygon (not just regular ones) always add to $360^{\circ}$.
- Imagine walking around the polygon: you turn through a full circle.
Each exterior angle of a regular hexagon is $60^\circ$, so each interior angle is $180^\circ-60^\circ=120^\circ$
The exterior angles of any polygon always add up to ______ degrees.
Walking around any polygon, you turn through exactly one full circle = 360°.
You've got it
- at a point $360^{\circ}$; on a line $180^{\circ}$; vertically opposite are equal
- parallel lines: corresponding (F) equal, alternate (Z) equal, co-interior (C) add to $180^{\circ}$
- polygon: interior sum $(n-2)\times 180^{\circ}$, exterior sum $360^{\circ}$