Angle facts, parallel lines and polygons · 角度事实、平行线与多边形
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| regular/ˈreɡjʊlə/ | 正的 | zhèng de |
| vertically opposite/ˈvɜːtɪkli ˈɒpəzɪt/ | 对顶角 | duì dǐng jiǎo |
| transversal/trænsˈvɜːsl/ | 截线 | jié xiàn |
| corresponding/ˌkɒrɪˈspɒndɪŋ/ | 同位角 | tóng wèi jiǎo |
| alternate/ɔːlˈtɜːnət/ | 内错角 | nèi cuò jiǎo |
| interior angles/ɪnˈtɪərɪə ˈæŋɡlz/ | 内角 | nèi jiǎo |
| polygon/ˈpɒlɪɡən/ | 多边形 | duō biān xíng |
| co-interior/kəʊ ɪnˈtɪərɪə/ | 同旁内角 | tóng páng nèi jiǎo |
| exterior angles/ekˈstɪərɪə ˈæŋɡlz/ | 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.
为什么蜂巢有六条边?
- 一个正六边形完美地铺贴而没有间隙——并且每个蜂房使用最少的蜡。
- 蜜蜂在数学家证明它之前数百万年就发现了这个。它背后的角的事实是你将在这里学的。
Parallel line and polygon lab · 平行线与多边形实验
Pick the angle rule that unlocks each diagram. · 选择能解开每个图形的角度规则。
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
一点处和一条线上的角的事实
- 一点处的角(angles at a point)加起来为 $360^{\circ}$。
- 一条直线上的角(angles on a straight line)加起来为 $180^{\circ}$。
- 对顶角(vertically opposite angles,两条线交叉处)相等。
一条线上的三个角:$x + 50^{\circ} + 70^{\circ} = 180^{\circ} \Rightarrow x = 60^{\circ}$。
在考试中给出理由。 单写"$60^{\circ}$"得不到分。你必须写"$60^{\circ}$,一条直线上的角加起来为 $180^{\circ}$。"
一个蜂巢用正六边形镶嵌平面
Three angles on a straight line are x, 50° and 70°. Find x (degrees). · 直线上有三个角，分别为 x、50° 和 70°。求 x（单位：度）。
x + 50 + 70 = 180, so x = 60°. · x + 50 + 70 = 180，所以 x = 60°。
Vertically opposite angles (where two lines cross) are always equal. · 对顶角（两条直线相交处）总是相等的。
When two lines cross, the angles opposite each other are equal. · 当两条直线相交时，相对的角相等。
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
平行线中的角
- 当一条横截线穿过两条平行线时:
- 同位角(corresponding,F 形)相等。
- 内错角(alternate,Z 形)相等。
- 同旁内角(co-interior,C 形)加起来为 $180^{\circ}$。
A co-interior angle to 110° (between parallel lines) is how many degrees? · 在平行线间，与 110° 互补的同旁内角是多少度？
Co-interior angles add to 180°: 180 − 110 = 70°. · 同旁内角之和为 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 $180^\circ$
多边形
- 对一个有 $n$ 条边的多边形:
- 一个正(regular)多边形所有的边和角都相等。
- 正六边形:外角 $= \dfrac{360}{6} = 60^{\circ}$,内角 $= 180 - 60 = 120^{\circ}$。
穿过两条平行线:同位角(F)和内错角(Z)相等;同旁内角(C)加起来为 $180^\circ$
Each interior angle of a regular hexagon is how many degrees? · 正六边形的每个内角是多少度？
Exterior = 360/6 = 60°, so interior = 180 − 60 = 120°. · 外角 = 360/6 = 60°，所以内角 = 180 − 60 = 120°。
The interior angles of a pentagon (5 sides) add up to how many degrees? · 五边形（5 条边）的内角之和是多少度？
(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 $180^\circ-60^\circ=120^\circ$
外角之和
- 任何多边形(不只是正的)的外角总是加起来为 $360^{\circ}$。
- 想象绕多边形走:你转过一整个圆。
一个正六边形的每个外角是 $60^\circ$,所以每个内角是 $180^\circ-60^\circ=120^\circ$
The exterior angles of any polygon always add up to ______ degrees. · 任何多边形的外角之和总是等于 ______ 度。
Walking around any polygon, you turn through exactly one full circle = 360°. · 绕任意多边形行走一圈，转过的角度正好是一整圈 = 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 $360^{\circ}$
你掌握了
- 一点处 $360^{\circ}$;一条线上 $180^{\circ}$;对顶角相等
- 平行线:同位角(F)相等,内错角(Z)相等,同旁内角(C)加起来为 $180^{\circ}$
- 多边形:内角之和 $(n-2)\times 180^{\circ}$,外角之和 $360^{\circ}$