📚 IGCSE Maths G2: Circle Theorems and Angle Properties | IGCSE 数学 G2：圆定理与角度性质
This topic brings together the angle properties that students meet across the IGCSE Maths syllabus, with a special focus on the circle theorems that are often tested in both Core and Extended papers.
本主题将 IGCSE 数学课程中各章节涉及的角性质综合在一起，并重点讲解核心与扩展试卷中常见的圆定理。
You will need to recall angle facts for straight lines, triangles, polygons, and circles, then apply them to unfamiliar diagrams. The key is not to memorise each theorem in isolation, but to see how one diagram can reveal several connected angle relationships.
你需要回顾直线、三角形、多边形和圆中的角事实，并将它们应用到陌生图形中。关键是不要孤立地死记每一条定理，而是要学会从同一个图形中发现多个相互关联的角关系。
1. Vertically Opposite and Parallel Lines | 对顶角与平行线
When two straight lines cross, the angles opposite each other are equal. These are called vertically opposite angles, and they are always equal in size.
当两条直线相交时，彼此相对的角相等。这些角叫做对顶角，它们总是大小相等。
If a transversal cuts two parallel lines, three useful facts appear: corresponding angles are equal, alternate interior angles are equal, and co-interior angles add up to 180°.
如果一条截线穿过两条平行线，会出现三个有用的事实：同位角相等、内错角相等、同旁内角之和为 180°。
| Corresponding angles | 同位角相等 | equal |
| Alternate interior angles | 内错角相等 | equal |
| Co-interior angles | 同旁内角互补 | sum to 180° |
2. Angles in Triangles | 三角形中的角
The sum of the interior angles of any triangle is always 180°. This fact is used constantly when finding missing angles in more complicated shapes.
任何三角形的内角和总是 180°。在更复杂的图形中求未知角时，这个事实会经常用到。
The exterior angle of a triangle is equal to the sum of the two opposite interior angles. This gives a quick shortcut in many IGCSE questions.
三角形的一个外角等于与它不相邻的两个内角之和。这在许多 IGCSE 题目中提供了一条快速解题的捷径。
A + B + C = 180°
三角形的内角和为 180°
3. Angles in Polygons | 多边形内角与外角
180°. For a regular polygon, each interior angle can be found by dividing this total by n.
对于任何有 n 条边的多边形，其内角和由公式 (n − 2) × 180° 给出。对于正多边形，每个内角可以用这个总和除以 n 来求得。
Sum of interior angles = (n − 2) × 180°
内角和 = (n − 2) × 180°
The sum of the exterior angles of any convex polygon is always 360°. This is true no matter how many sides the polygon has, and it often provides a much simpler method for regular polygon problems.
任何凸多边形的外角和总是 360°。无论多边形有多少条边，这一点都成立，并且它常常为正多边形问题提供更简单的方法。
4. Circle Terminology | 圆的基本术语
Before using circle theorems, you must recognise the parts of a circle: radius, diameter, chord, tangent, arc, sector, and segment. A chord is a line segment joining two points on the circumference, while a tangent touches the circle at exactly one point.
在使用圆定理之前，你必须认识圆的各个部分：半径、直径、弦、切线、弧、扇形和弓形。弦是连接圆上两点的线段，而切线与圆只有一个接触点。
- Radius: distance from centre to circumference
- 半径：圆心到圆周的距离
- Diameter: a chord passing through the centre
- 直径：经过圆心的弦
- Tangent: a line that touches the circle once
- 切线：与圆只有一个接触点的直线
- Arc: a part of the circumference
- 弧：圆周的一部分
5. Central and Inscribed Angles | 圆心角与圆周角
The central angle is the angle at the centre of the circle, formed by two radii. The inscribed angle is the angle at the circumference, formed by two chords that share an endpoint.
圆心角是由两条半径在圆心处形成的角。圆周角是由两条弦在圆周处形成的角，这两条弦共用一个端点。
If both angles stand on the same arc, then the central angle is exactly twice the inscribed angle. This is one of the most frequently used circle theorems in IGCSE exams.
如果两个角都对着同一条弧，那么圆心角恰好是圆周角的两倍。这是 IGCSE 考试中最常用的圆定理之一。
Central angle = 2 × Inscribed angle
圆心角 = 2 × 圆周角
6. Angles in the Same Segment | 同弧上的圆周角
All inscribed angles that stand on the same arc, and are in the same segment of the circle, are equal to one another.
所有对着同一条弧且位于同一弓形内的圆周角都彼此相等。
This theorem is very useful when a question gives one angle at the circumference and asks you to find another angle that appears in a different part of the diagram but stands on the same chord or arc.
当题目给出圆周上的一个角，并要求你求另一个角，而这个角在图形不同位置却对着同一条弦或弧时，这一定理非常有用。
7. Angle in a Semicircle | 半圆内角
An angle inscribed in a semicircle is always a right angle. In other words, if one side of an inscribed triangle is the diameter of the circle, then the angle opposite that diameter is 90°.
半圆内的圆周角总是直角。换句话说，如果圆内接三角形的一条边是圆的直径，那么这条直径所对的圆周角就是 90°。
This theorem is regularly hidden inside IGCSE problems. Look for a diameter and a point on the circumference, and you can immediately mark a right angle.
这一定理经常隐藏在 IGCSE 题目中。只要看到直径和圆周上的一点，你就可以立刻标出一个直角。
8. Cyclic Quadrilaterals | 内接四边形
A cyclic quadrilateral is a quadrilateral whose four vertices all lie on the circumference of a circle. In a cyclic quadrilateral, opposite angles are supplementary, meaning they add up to 180°.
内接四边形是指四个顶点都在同一个圆上的四边形。在内接四边形中，对角互补，即对角之和为 180°。
∠A + ∠C = 180° and ∠B + ∠D = 180°
∠A + ∠C = 180°，∠B + ∠D = 180°
This gives a powerful way to find an unknown angle when three other angles of the cyclic quadrilateral are known, or when one opposite pair is linked by algebra.
当内接四边形的其他三个角已知，或者一对对角之间存在代数关系时，这是一个求未知角的强大方法。
9. Tangent and Radius | 切线与半径
A tangent to a circle is always perpendicular to the radius drawn to the point of contact. This creates a right angle at the point where the tangent touches the circle.
圆的切线总是垂直于经过切点的半径。这在切线与圆接触的点处形成一个直角。
In many IGCSE questions, this theorem is combined with Pythagoras' theorem because the radius, tangent, and a line from the centre to an external point form a right-angled triangle.
在许多 IGCSE 题目中，这一定理常与毕达哥拉斯定理结合使用，因为半径、切线和从圆心到外部点的连线构成一个直角三角形。
Also, from the same external point, the two tangents drawn to a circle are equal in length.
此外，从同一个外部点引出的两条切线长度相等。
10. Chord and Perpendicular Bisector | 弦与垂直平分线
The perpendicular bisector of a chord passes through the centre of the circle. Equally, a line from the centre perpendicular to a chord bisects that chord.
弦的垂直平分线经过圆心。同样，从圆心垂直于弦的线会平分这条弦。
This theorem is useful when you are given information about a chord and need to find the distance from the centre to the chord. A right-angled triangle can be formed with the radius, half the chord, and the perpendicular distance.
当题目给出有关弦的信息，需要求圆心到弦的距离时，这一定理很有用。半径、弦的一半和垂直距离可以构成一个直角三角形。
11. Alternate Segment Theorem | 交错弦切角定理
The alternate segment theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle subtended by that chord in the alternate segment of the circle.
交错弦切角定理指出，切线与经过切点的弦之间的夹角等于该弦在圆的另一弓形中所对的圆周角。
This is an Extended-level theorem, but it appears regularly in IGCSE past papers. Look for a tangent and a chord, and mark the angle in the opposite segment as equal.
这是一条扩展级别的定理，但经常出现在 IGCSE 历年试题中。看到切线和弦时，就可以把另一侧弓形中的角标记为相等。
12. Exam Tips and Common Errors | 考试提示与常见错误
When answering circle theorem questions, always state which theorem you are using. Even if your final answer is correct, IGCSE examiners often award method marks for a clear justification.
在解答圆定理题目时，始终说明你使用的是哪一条定理。即使最终答案正确，IGCSE 考官也通常会为清晰的证明过程给方法分。
- Common error: assuming any angle at the centre is double any angle at the circumference, even when they do not stand on the same arc.
- 常见错误：以为任何圆心角都是任何圆周角的两倍，即使它们并不对着同一条弧。
- Common error: confusing co-interior angles with alternate interior angles in parallel line problems.
- 常见错误：在平行线问题中混淆同旁内角与内错角。
- Common error: forgetting that opposite angles in a cyclic quadrilateral are supplementary only when all four vertices lie on the circle.
- 常见错误：忘记只有当四个顶点都在圆上时，内接四边形的对角才互补。
Always re-read the question to check whether the shape is drawn inside a circle, whether lines are parallel, and whether a line is a tangent before applying a theorem.
在应用定理之前，一定要重新审题，确认图形是否内接于圆、直线是否平行、线段是否为切线。
Published by TutorHao | Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图