IGCSE Mathematics: Lines & Angles | IGCSE数学：线与角

📚 IGCSE Mathematics: Lines & Angles | IGCSE数学：线与角

Lines and angles form the foundation of all geometry. In IGCSE Mathematics, you are expected to understand the properties of straight lines, parallel lines, and the angles formed when they are intersected by a transversal. This article covers every key concept you need, with clear explanations and exam-style examples.

线与角是全部几何的基础。在 IGCSE 数学中，你需要理解直线、平行线以及被截线相交时所形成角的所有性质。本文将涵盖你需要的每一个关键概念，并提供清晰的解释和考试风格的例题。

1. Basic Angle Definitions | 基本角的定义

An angle is formed when two rays meet at a common endpoint called the vertex. Angles are measured in degrees (°). A full rotation is 360°, a half rotation is 180°, and a quarter rotation is 90°.

角是由两条射线在公共端点（称为顶点）处相交而形成的。角用度数（°）来度量。一整圈是 360°，半圈是 180°，四分之一圈是 90°。

Acute angle: between 0° and 90° | 锐角：在 0° 和 90° 之间
Right angle: exactly 90° | 直角：正好 90°
Obtuse angle: between 90° and 180° | 钝角：在 90° 和 180° 之间
Straight angle: exactly 180° | 平角：正好 180°
Reflex angle: between 180° and 360° | 优角（反射角）：在 180° 和 360° 之间

When naming an angle, use three letters with the vertex in the middle, for example ∠ABC means the angle at point B formed by BA and BC.

在命名角时，使用三个字母，顶点放在中间，例如 ∠ABC 表示由 BA 和 BC 在点 B 处形成的角。

2. Angle Relationships on a Straight Line | 直线上的角关系

Angles on a straight line add up to 180°. This is one of the most frequently used rules in angle problems.

一条直线上的角之和为 180°。这是角度问题中最常用的规则之一。

a + b + c = 180°

If three angles, say 40°, 60° and x°, lie on a straight line, then x = 180 − 40 − 60 = 80°.

如果三个角，比如 40°、60° 和 x°，位于一条直线上，那么 x = 180 − 40 − 60 = 80°。

Angles around a point add up to 360°. If four angles around a point are 90°, 70°, 100° and y°, then y = 360 − 90 − 70 − 100 = 100°.

一个点周围的角之和为 360°。如果围绕一点有四个角分别是 90°、70°、100° 和 y°，那么 y = 360 − 90 − 70 − 100 = 100°。

Vertically opposite angles are equal. When two straight lines cross, the angles opposite each other are always equal.

对顶角相等。当两条直线相交时，彼此相对的角总是相等。

3. Perpendicular and Parallel Lines | 垂直与平行线

Perpendicular lines meet at a right angle (90°). Parallel lines are always the same distance apart and never meet, no matter how far they are extended.

垂线以直角（90°）相交。平行线始终保持相同的距离，无论延伸多远都不会相交。

When a transversal (a line that crosses two or more parallel lines) intersects parallel lines, several special angle relationships appear.

当一条截线（一条穿过两条或更多平行线的直线）与平行线相交时，会出现几种特殊的角关系。

Corresponding angles are equal | 同位角相等

Alternate angles are equal | 内错角相等

Interior angles (co-interior) sum to 180° | 同旁内角之和为 180°

In diagrams, corresponding angles are often marked with the same symbol (like a small ‘f’ shape). Alternate angles form a ‘z’ shape, and co-interior angles form a ‘c’ shape.

在图形中，同位角通常用相同的符号标记（像一个小写的 ‘f’ 形）。内错角形成 ‘z’ 字形，同旁内角形成 ‘c’ 字形。

4. Triangles | 三角形

The interior angles of any triangle add up to 180°. This rule is used constantly in angle calculations.

任何三角形的内角之和为 180°。这个规则在角度计算中经常使用。

a + b + c = 180°

An equilateral triangle has three equal sides and three equal angles, each 60°. An isosceles triangle has two equal sides and two equal base angles. A scalene triangle has all sides and angles different.

等边三角形有三条相等边和三个相等角，每个角为 60°。等腰三角形有两条相等边和两个相等的底角。不等边三角形的所有边和角都不相等。

An exterior angle of a triangle is equal to the sum of the two opposite interior angles. For example, if a triangle has interior angles 50° and 70°, the exterior angle at the third vertex is 50 + 70 = 120°.

三角形的外角等于与它不相邻的两个内角之和。例如，如果一个三角形的内角为 50° 和 70°，那么第三个顶点处的外角为 50 + 70 = 120°。

5. Quadrilaterals | 四边形

The interior angles of any quadrilateral add up to 360°. You can see this by dividing the quadrilateral into two triangles.

任何四边形的内角之和为 360°。你可以通过将四边形分成两个三角形来看出这一点。

a + b + c + d = 360°

Shape	Properties	图形	性质
Square	4 equal sides, 4 right angles, opposite sides parallel	正方形	4 条等边，4 个直角，对边平行
Rectangle	Opposite sides equal and parallel, 4 right angles	矩形	对边相等且平行，4 个直角
Parallelogram	Opposite sides equal and parallel, opposite angles equal	平行四边形	对边相等且平行，对角相等
Rhombus	4 equal sides, opposite angles equal, diagonals meet at right angles	菱形	4 条等边，对角相等，对角线垂直相交
Trapezium	One pair of parallel sides	梯形	一组对边平行
Kite	Two pairs of adjacent equal sides, one pair of equal opposite angles	筝形	两组邻边相等，一对对角相等

6. Polygons | 多边形

A polygon is a closed shape with straight sides. For a polygon with n sides, the sum of interior angles is given by:

多边形是由直边围成的封闭图形。对于有 n 条边的多边形，其内角和由下式给出：

Sum of interior angles = (n − 2) × 180°

For a regular polygon, all sides and all angles are equal. Each interior angle is calculated by dividing the total sum by n.

对于正多边形，所有边和所有角都相等。每个内角通过将总和除以 n 来计算。

Each interior angle of a regular n-gon = (n − 2) × 180° ÷ n

The exterior angle of a regular polygon can also be found. The sum of all exterior angles of any polygon is always 360°. Therefore:

正多边形的外角也可以求出。任何多边形的所有外角之和始终为 360°。因此：

Each exterior angle of a regular n-gon = 360° ÷ n

For example, a regular hexagon (n = 6) has interior angles of (6 − 2) × 180 ÷ 6 = 120°, and exterior angles of 360 ÷ 6 = 60°.

例如，正六边形（n = 6）的内角为 (6 − 2) × 180 ÷ 6 = 120°，外角为 360 ÷ 6 = 60°。

7. Angle Problems Using Algebra | 用代数解决角问题

In IGCSE exams, you will often need to set up equations involving angles. Start by writing down the relevant angle rule, then substitute the given expressions.

在 IGCSE 考试中，你通常需要建立涉及角的方程。先写下相关的角规则，然后代入给定的表达式。

Example: Two parallel lines are cut by a transversal. One interior angle is (2x + 10°) and the co-interior angle is (3x + 20°). Find x.

示例：两条平行线被一条截线所截。其中一个同旁内角为 (2x + 10°)，另一个同旁内角为 (3x + 20°)。求 x。

Since co-interior angles sum to 180°, we write:

因为同旁内角之和为 180°，我们写出：

(2x + 10) + (3x + 20) = 180

5x + 30 = 180

5x = 150

x = 30

Always check your answer by substituting back. 2(30) + 10 = 70°, and 3(30) + 20 = 110°. Since 70 + 110 = 180°, the answer is correct.

始终通过代回原式来检查答案。2(30) + 10 = 70°，且 3(30) + 20 = 110°。因为 70 + 110 = 180°，答案正确。

8. Bearings | 方位角

A bearing is an angle measured clockwise from north, written as a three-digit number. For example, north-east is 045°, due east is 090°, and due south is 180°.

方位角是从正北方向顺时针测量的角，用三位数表示。例如，东北方向是 045°，正东是 090°，正南是 180°。

Bearings are essential in navigation and map-reading questions. To find a bearing from one point to another, measure the angle from north at the starting point, moving clockwise to the line connecting the two points.

方位角在导航和读图题中至关重要。要求从一点到另一点的方位角时，在起点处从正北开始顺时针测量到连接两点的直线所成的角。

If an angle is given as a normal angle (measured anticlockwise from north), the bearing is calculated by subtracting the angle from 360°.

如果给出的是普通角（从正北逆时针测量），则方位角通过用 360° 减去该角来计算。

9. Common Exam Mistakes | 常见考试错误

Students often confuse corresponding and alternate angles. Remember: corresponding angles occupy the same relative position at each intersection, while alternate angles are on opposite sides of the transversal and lie between the parallel lines.

学生经常混淆同位角和内错角。记住：同位角在两个交点处占据相同的相对位置，而内错角位于截线的两侧且在两平行线之间。

Another common mistake is forgetting that angles in a triangle sum to 180°, not 360°. Always check whether you are dealing with a triangle or a quadrilateral.

另一个常见错误是忘记三角形内角和为 180° 而不是 360°。始终检查你面对的是三角形还是四边形。

Also, when solving for x, do not stop once you find x. The question may ask for a specific angle, so be sure to substitute x back into the original expression.

此外，在求解 x 时，不要找到 x 就停止。题目可能要求某个具体角，所以一定要将 x 代回原表达式。

10. Revision Checklist | 复习清单

Use this checklist to confirm you are ready for your exam:

使用这份清单来确认你已经为考试做好准备：

I can classify acute, right, obtuse, straight and reflex angles | 我能区分锐角、直角、钝角、平角和优角
I know angles on a straight line sum to 180° and angles around a point sum to 360° | 我知道直线上的角之和为 180°，一个点周围的角之和为 360°
I can identify corresponding, alternate and co-interior angles | 我能识别同位角、内错角和同旁内角
I can use the exterior angle rule for triangles | 我能使用三角形的外角规则
I can calculate interior and exterior angles of regular polygons | 我能计算正多边形的内角和外角
I can set up and solve angle equations with algebra | 我能建立并求解含代数的角方程
I can read and write bearings correctly | 我能正确读取和书写方位角

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IGCSE Mathematics: Lines & Angles | IGCSE数学：线与角