📚 Angles, Polygons and Parallel Lines | 角、多边形与平行线
In IGCSE Mathematics, angle geometry is one of the most frequently assessed topics. It connects line properties, triangle facts, polygon formulas and parallel line rules. Many candidates can state individual rules but lose marks when they have to combine several ideas in one diagram. This revision guide brings together the essential angle facts and shows how to apply them step by step.
在 IGCSE 数学中，角的几何是最常考查的主题之一。它将直线性质、三角形事实、多边形公式和平行线规则联系起来。许多考生能够说出单个规则，但当需要在一个图形中综合运用多个知识点时却容易失分。本复习指南汇总了核心角的事实，并展示如何逐步应用它们。
1. Basic Angle Facts | 角的基本事实
Angles are measured in degrees. A full turn is 360°, a straight line is 180°, and a right angle is 90°. You also need to recognise acute angles (less than 90°), obtuse angles (between 90° and 180°), reflex angles (between 180° and 360°), complementary angles (sum 90°) and supplementary angles (sum 180°).
角以度为单位测量。一整圈是 360°，直线是 180°，直角是 90°。你还需要识别锐角（小于 90°）、钝角（在 90° 与 180° 之间）、优角（在 180° 与 360° 之间）、余角（和为 90°）和补角（和为 180°）。
When two straight lines intersect, the opposite angles are equal. These are called vertically opposite angles. This fact is often used with parallel lines or within triangles, so it is important to identify the intersection point clearly.
当两条直线相交时，对顶角相等。这些角称为对顶角。这一事实经常与平行线或三角形结合使用，因此清晰地识别交点是十分重要的。
2. Angles on a Straight Line and Around a Point | 直线上的角与一点周围的角
If several angles meet at a point on a straight line, their sum is 180°. This is the straight-line rule: a + b + c = 180°. If angles meet around a point with no gaps, their total is 360°.
如果几个角在一条直线上某点相遇，它们的和是 180°。这是直线规则：a + b + c = 180°。如果角在一个点周围无空隙地相遇，它们的总和是 360°。
These two rules are very common in IGCSE papers. You may be given one or two unknown angles on a straight line and asked to form an equation. Always check whether the diagram shows a straight line or a complete turn before adding the angles, because using the wrong total is a frequent source of error.
这两条规则在 IGCSE 试卷中非常常见。你可能会得到直线上的一两个未知角，并被要求列出方程。在将角度相加之前，始终检查图形显示的是直线还是一整圈，因为使用错误的总和是一个常见的错误来源。
3. Parallel Lines and Transversals | 平行线与截线
When a transversal crosses two parallel lines, three key angle relationships are formed. Corresponding angles are equal; they lie on the same side of the transversal and in matching positions. Alternate angles are equal; they lie between the parallel lines on opposite sides of the transversal. Co-interior angles are supplementary; they lie between the parallel lines on the same side of the transversal and add up to 180°.
当一条截线与两条平行线相交时，会形成三种关键的角关系。同位角相等；它们位于截线的同一侧且位置对应。内错角相等；它们位于两条平行线之间且在截线的两侧。同旁内角互补；它们位于两条平行线之间且在截线的同一侧，相加为 180°。
To use these rules correctly, first identify the parallel lines and the transversal. Then select the angle relationship that matches the positions. Drawing arrows on the parallel lines can help you see which angles correspond and avoid matching the wrong pair.
要正确使用这些规则，首先确定平行线和截线。然后选择与位置匹配的角关系。在平行线上画箭头可以帮助你看出哪些角对应，避免匹配错误的角对。
4. Angle Properties of Triangles | 三角形的角性质
The interior angles of any triangle add up to 180°. This means a + b + c = 180° for a triangle with angles a, b and c. An exterior angle of a triangle equals the sum of the two interior opposite angles.
任何三角形的内角和为 180°。这意味着对于角为 a、b 和 c 的三角形，a + b + c = 180°。三角形的一个外角等于两个不相邻内角之和。
Special triangles have extra angle information. An equilateral triangle has all angles 60°. An isosceles triangle has two equal sides and the angles opposite those sides are equal. A right-angled triangle has one 90° angle, so the other two angles are complementary.
特殊三角形有额外的角度信息。等边三角形的所有角都是 60°。等腰三角形有两条相等的边，且等边所对的角相等。直角三角形有一个 90° 角，因此另外两个角互余。
5. Angle Properties of Quadrilaterals | 四边形的角性质
The sum of the interior angles of any quadrilateral is 360°. This follows from splitting the quadrilateral into two triangles: 2 × 180° = 360°. You should also know the angle properties of specific quadrilaterals such as squares, rectangles, parallelograms, rhombuses, trapeziums and kites.
任何四边形的内角和为 360°。这是因为将四边形分成两个三角形：2 × 180° = 360°。你还应了解特殊四边形的角性质，如正方形、长方形、平行四边形、菱形、梯形和风筝形。
A parallelogram has opposite angles equal and adjacent angles supplementary. In a kite, one pair of opposite angles are equal. In a trapezium, angles on each leg are supplementary when the trapezium has one pair of parallel sides.
平行四边形的对角相等，邻角互补。在风筝形中，有一组对角相等。在梯形中，当梯形有一组对边平行时，每条腰上的两个角互补。
6. Interior Angles of Polygons | 多边形的内角
For any polygon with n sides, the sum of the interior angles is (n – 2) × 180°. This formula comes from dividing the polygon into n – 2 triangles. For example, a hexagon has n = 6, so the interior angle sum is (6 – 2) × 180° = 720°.
对于任何有 n 条边的多边形，内角和为 (n – 2) × 180°。这个公式来源于将多边形分成 n – 2 个三角形。例如，六边形有 n = 6，因此内角和为 (6 – 2) × 180° = 720°。
If the polygon is regular, all interior angles are equal. Then one interior angle is (n – 2) × 180° ÷ n. For example, each interior angle of a regular pentagon is (5 – 2) × 180° ÷ 5 = 108°.
如果多边形是正多边形，所有内角相等。那么一个内角为 (n – 2) × 180° ÷ n。例如，正五边形的每个内角为 (5 – 2) × 180° ÷ 5 = 108°。
7. Exterior Angles of Polygons | 多边形的外角
The exterior angles of any convex polygon add up to 360°. At each vertex, the interior angle and the corresponding exterior angle lie on a straight line, so they add up to 180°.
任何凸多边形的外角和为 360°。在每个顶点处，内角与相应的外角位于同一直线上，因此它们相加为 180°。
For a regular polygon, all exterior angles are equal, so one exterior angle is 360° ÷ n. This is often the quickest way to find the number of sides when you are given an exterior angle. For example, if a regular polygon has an exterior angle of 24°, then n = 360° ÷ 24° = 15.
对于正多边形，所有外角相等，因此一个外角为 360° ÷ n。当你已知外角时，这通常是求边数最快的方法。例如，如果一个正多边形的外角为 24°，那么 n = 360° ÷ 24° = 15。
8. Regular Polygons and Symmetry | 正多边形与对称性
A regular polygon has all sides equal and all angles equal. It has the same number of lines of symmetry as its number of sides. It also has rotational symmetry of order n about its centre.
正多边形的所有边相等，所有角相等。它的对称轴数量与其边数相同。它还关于其中心具有 n 阶旋转对称性。
These symmetry properties are sometimes tested alongside angle calculations. For instance, the interior angle of a regular octagon can be found by the formula, but you can also check it by considering the symmetry of the shape. Knowing both angle and symmetry facts gives you a fuller understanding of regular polygons.
这些对称性质有时与角度计算一起考查。例如，正八边形的内角可以用公式求出，但你也可以通过考虑图形的对称性来验证。同时掌握角度和对称性的事实能让你更全面地理解正多边形。
9. Worked Examples and Exam Techniques | 例题与解题技巧
Example: A triangle has angles (2x + 10)°, (3x – 20)° and (x + 30)°. Find x and state the size of each angle. Solution: The sum is 180°, so (2x + 10) + (3x – 20) + (x + 30) = 180. That gives 6x + 20 = 180, so 6x = 160 and x = 26.7° approximately. The angles are then 63.3°, 60° and 56.7°.
例题：一个三角形的角分别为 (2x + 10)°、(3x – 20)° 和 (x +
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