📚 G2 Angle Properties and Parallel Lines | G2 角度性质与平行线
In IGCSE Mathematics, angle properties form the foundation of many geometry problems. This article reviews the essential angle vocabulary, the rules for angles on straight lines and around points, the special angles created by parallel lines, and the angle facts for triangles and polygons. Mastering these rules will help you solve both short-answer questions and multi-step geometric proofs with confidence.
在 IGCSE 数学中，角度性质是许多几何问题的基础。本文系统复习角度基本词汇、直线与同一点周围的角度关系、平行线产生的特殊角度，以及三角形与多边形的角度定理。掌握这些规则，能够帮助你自信地解答简答题和多步骤几何证明题。
1. Angle Vocabulary and Notation | 角度词汇与记号
An angle is formed when two rays meet at a common endpoint. The common endpoint is called the vertex, and the two rays are called the arms of the angle. Angles are usually measured in degrees and the symbol for degrees is °. A full rotation is 360°.
角是由两条射线在公共端点处相交而形成的图形。这个公共端点称为顶点，两条射线称为角的边。角通常用度来度量，度的符号是 °，一整圈旋转是 360°。
To name an angle with three letters, such as ∠ABC, the middle letter represents the vertex. Sometimes a single lower-case letter or a Greek letter such as θ is used.
用三个字母表示角时，例如 ∠ABC，中间字母表示顶点。有时也可以用单个小写字母或希腊字母如 θ 来表示一个角。
| Type of angle / 角类型 | Measure / 范围 | Key fact / 要点 |
| Acute angle / 锐角 | 0° < θ < 90° | Smaller than a right angle / 小于直角 |
| Right angle / 直角 | θ = 90° | Often marked with a square / 常画小方框表示 |
| Obtuse angle / 钝角 | 90° < θ < 180° | Larger than a right angle / 大于直角 |
| Straight angle / 平角 | θ = 180° | Forms a straight line / 形成一条直线 |
| Reflex angle / 优角 | 180° < θ < 360° | Greater than a straight angle / 大于平角 |
2. Angles on a Straight Line and Around a Point | 直线上的角与同一点周围的角
ne. For example, if three angles a, b and c lie on a straight line, then a + b + c = 180°.
当多个角在同一条直线上时，它们的度数和总是 180°。这称为“平角等于 180°”。例如，如果三个角 a、b、c 在同一直线上，则 a + b + c = 180°。
a + b = 180°
When angles meet at a single point, the total around that point is 360°. This fact is useful when solving problems involving bearings, pie charts, or crossing lines.
当多个角围绕同一个点分布时，围绕该点的角的总和是 360°。这个结论在方向角、扇形统计图以及直线相交问题中非常有用。
p + q + r + s = 360°
3. Vertically Opposite Angles | 对顶角
When two straight lines intersect, they form four angles. The angles directly opposite each other are called vertically opposite angles. Vertically opposite angles are equal.
当两条直线相交时，它们形成四个角。正相对的两个角称为对顶角。对顶角相等。
x = y
For example, if two lines cross and one angle is 35°, then the vertically opposite angle is also 35°. The adjacent angles are each 145°, because they form a straight line and must add to 180°.
例如，两条直线相交，若其中一个角是 35°，那么它的对顶角也是 35°。相邻的两个角各为 145°，因为它们在同一条直线上，和必须为 180°。
4. Parallel Lines and a Transversal | 平行线与截线
Parallel lines are lines in the same plane that never meet. In diagrams, parallel lines are usually marked with arrowheads. A line that crosses two or more parallel lines is called a transversal.
平行线是在同一平面内永远不会相交的直线。在图中，平行线通常用小箭头标记。穿过两条或更多平行线的直线称为截线。
When a transversal intersects a pair of parallel lines, it creates eight angles. These angles have special relationships: corresponding angles, alternate angles, and co-interior angles. We study each relationship below.
当一条截线与一对平行线相交时，会形成八个角。这些角之间具有特殊关系：同位角、内错角和同旁内角。下面我们逐一学习这些关系。
5. Corresponding Angles | 同位角
Corresponding angles are on the same side of the transversal and in matching positions at the two intersections. They form an "F" pattern. Corresponding angles are equal.
同位角位于截线的同一侧，并且在两个交点处的对应位置上。它们构成类似字母“F”的图形。同位角相等。
corresponding angle = corresponding angle
For example, if a transversal makes a 70° angle with the first parallel line, the corresponding angle with the second parallel line is also 70°.
例如，如果截线与第一条平行线成 70°角，那么它与第二条平行线形成的同位角也是 70°。
6. Alternate Angles | 内错角
Alternate angles are on opposite sides of the transversal and between the two parallel lines. They form a "Z" pattern. Alternate angles are equal.
内错角位于截线的两侧，并且夹在两条平行线之间。它们构成类似字母“Z”的图形。内错角相等。
alternate angle = alternate angle
A common exam question shows two parallel lines with a diagonal transversal; the highest angle on one side is equal to the lowest angle on the other side. Look for the "Z" shape to identify alternate angles.
常见的考试题会画出两条平行线和一条斜截线；一侧上方的角等于另一侧下方的角。寻找“Z”字形可以快速识别内错角。
7. Co-Interior Angles | 同旁内角
Co-interior angles, also called allied angles, are on the same side of the transversal and between the two parallel lines. They form a "C" pattern. Co-interior angles add up to 180°.
同旁内角，也称为平行线同侧内角，位于截线的同一侧且夹在两条平行线之间。它们构成类似字母“C”的图形。同旁内角互补，和为 180°。
co-interior angles sum to 180°
For example, if one co-interior angle is 110°, the other must be 70°. Adjacent interior angles on the same side of a transversal are supplementary.
例如，若一个同旁内角为 110°，另一个必须是 70°。截线同侧的两个平行线内角互补。
8. Using Angle Rules to Prove Lines Are Parallel | 利用角度定理判定平行线
The three rules above can be used in reverse. If a pair of corresponding angles are equal, then the lines are parallel. If a pair of alternate angles are equal, then the lines are parallel. If a pair of co-interior angles add to 180°, then the lines are parallel.
上述三条定理也可以反向使用。若一对同位角相等，则两直线平行；若一对内错角相等，则两直线平行；若一对同旁内角的和为 180°，则两直线平行。
If ∠a = ∠b, then l ∥ m
In proofs, you should state the reason clearly, for example: "Because alternate angles are equal, line AB is parallel to line CD." Always look for arrowhead markings in the diagram before making this conclusion.
在证明题中，你需要清楚地写出理由，例如：“因为内错角相等，所以直线 AB 平行于直线 CD。”在作出结论前，一定要先看图中是否有箭头标记。
9. Angles in Triangles | 三角形中的角
The interior angles of any triangle add to 180°. This is one of the most important angle facts in IGCSE Mathematics.
任意三角形的内角和都是 180°。这是 IGCSE 数学中最重要的角度定理之一。
a + b + c = 180°
There are also two useful facts: an exterior angle of a triangle is equal to the sum of the two opposite interior angles; in an isosceles triangle, the base angles are equal.
另外还有两条重要结论：三角形的一个外角等于它不相邻的两个内角之和；等腰三角形的两个底角相等。
exterior angle = opposite interior angle + opposite interior anglePublished by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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