Unit 06 Angles: Parallel Lines, Polygons and Reasoning | 第六单元角度：平行线、多边形与推理

📚 Unit 06 Angles: Parallel Lines, Polygons and Reasoning | 第六单元角度：平行线、多边形与推理

This revision guide covers the essential angle rules tested in GCSE Higher Unit 6 (pages 124-142). You will learn how to identify angle relationships on parallel lines, calculate interior and exterior angles of polygons, and use deductive reasoning to solve multi-step problems.

本复习指南涵盖 GCSE 高等数学第六单元（第124-142页）考查的核心角度规则。你将学习如何识别平行线上的角度关系，计算多边形的内角与外角，并运用演绎推理解决多步骤问题。

1. Angle Basics and Notation | 角度基础与标记

An angle is formed when two lines meet at a point. We label angles using three capital letters, with the middle letter at the vertex, or by writing an angle sign followed by a single letter.

当两条直线在一点相交时形成角。我们用三个大写字母标记角，中间字母表示顶点，也可以写成角符号后加一个字母。

The size of an angle is measured in degrees (°). A full turn is 360°, a half turn is 180°, and a quarter turn is 90°. Knowing these key turns helps you check whether an answer is reasonable.

角的大小以度（°）为单位。一整圈为360°，半圈为180°，四分之一圈为90°。掌握这些关键转量有助于你检查答案是否合理。

Acute angle: less than 90° | 锐角：小于90°
Right angle: exactly 90° | 直角：恰好90°
Obtuse angle: between 90° and 180° | 钝角：90°到180°之间
Reflex angle: between 180° and 360° | 优角：180°到360°之间

2. Angles at a Point and on a Straight Line | 同一点与直线上的角

Angles around a single point always add up to 360°. This is because they complete a full turn. If several angles meet at a point, you can find a missing angle by subtracting the known angles from 360°.

围绕同一点的角之和始终为360°。这是因为它们形成一整圈。如果几个角相交于一点，你可以用360°减去已知角来求未知角。

Angles on a straight line add up to 180°. When two or more angles lie along a straight edge, their sum is a half turn. This rule is often used together with parallel line angle facts.

同一直线上的角之和为180°。当两个或更多角沿直线排列时，它们的和是半圈。这条规则常与平行线角的相关事实一起使用。

a + b + c = 180° (straight line) | a + b + c = 180°（直线）

3. Vertically Opposite Angles | 对顶角

When two straight lines cross, the opposite angles are called vertically opposite angles. Vertically opposite angles are always equal. This is because each pair lies on the same two straight lines, so both must have the same supplement.

当两条直线相交时，相对的角称为对顶角。对顶角始终相等。这是因为每对角都位于相同的两条直线上，所以它们必须具有相同的补角。

For example, if two lines intersect and one angle is 65°, the angle directly across from it is also 65°. The other two angles, which are also vertically opposite, each measure 115° because they lie on a straight line with 65°.

例如，如果两条直线相交，其中一个角为65°，那么它正对面的角也是65°。另外两个角也是对顶角，且各为115°，因为它们与65°角在同一直线上。

Given angle \| 已知角	65°
Vertically opposite angle \| 对顶角	65°
Adjacent angle on a straight line \| 直线上的邻角	115°

4. Parallel Lines: Corresponding, Alternate and Co-interior Angles | 平行线：同位角、内错角与同旁内角

When a transversal crosses two parallel lines, special angle pairs are formed. Recognising these pairs is one of the most important skills in Unit 6.

当一条截线穿过两条平行线时，会形成特殊的角对。识别这些角对是第六单元最重要的技能之一。

Corresponding angles are in the same position at each intersection. They are equal. | 同位角位于每个交点的相同位置。它们相等。
Alternate angles are on opposite sides of the transversal but inside the parallel lines. They are equal. | 内错角位于截线的两侧但在平行线内侧。它们相等。
Co-interior angles are on the same side of the transversal and inside the parallel lines. They add up to 180°. | 同旁内角位于截线同侧且在平行线内侧。它们的和为180°。

You can quickly identify these angles by looking for the characteristic F, Z and C shapes. Corresponding angles form an F, alternate angles form a Z, and co-interior angles form a C.

你可以通过观察典型的F形、Z形和C形快速识别这些角。同位角形成F形，内错角形成Z形，同旁内角形成C形。

Corresponding: ∠a = ∠b | 同位角：∠a = ∠b

Alternate: ∠c = ∠d | 内错角：∠c = ∠d

Co-interior: ∠e + ∠f = 180° | 同旁内角：∠e + ∠f = 180°

5. Triangles and Angle Sum | 三角形与内角和

The angles inside any triangle always add up to 180°. This fact is used constantly in multi-step angle problems, especially when combined with parallel lines or polygons.

任何三角形内部的角之和始终为180°。这一事实在多步骤角度问题中经常被使用，特别是与平行线或多边形结合时。

An equilateral triangle has three equal angles of 60°. An isosceles triangle has two equal angles at the base. Knowing the triangle type helps you find unknown angles quickly.

等边三角形的三个角相等，均为60°。等腰三角形的两个底角相等。了解三角形类型有助于你快速求出未知角。

There are also two useful angle facts involving triangles: the exterior angle of a triangle is equal to the sum of the two opposite interior angles, and the sum of all interior angles is 180°.

关于三角形还有两个有用的角事实：三角形的外角等于两个不相邻内角之和，且所有内角之和为180°。

∠A + ∠B + ∠C = 180° | ∠A + ∠B + ∠C = 180°

6. Quadrilaterals and Angle Sum | 四边形与内角和

Any quadrilateral can be split into two triangles by drawing one diagonal. Since each triangle has an angle sum of 180°, the total interior angle sum of a quadrilateral is 2 × 180° = 360°.

任意四边形都可以通过画一条对角线分成两个三角形。由于每个三角形的内角和为180°，四边形的内角总和为2 × 180° = 360°。

This rule applies to all quadrilaterals, including squares, rectangles, parallelograms, rhombuses, trapeziums and irregular four-sided shapes. You can also use it to check your work.

这条规则适用于所有四边形，包括正方形、长方形、平行四边形、菱形、梯形和不规则四边形。你也可以用它来检查你的答案。

In some quadrilaterals, angle properties are even more specific. For example, in a parallelogram opposite angles are equal and adjacent angles add up to 180° because they are co-interior on parallel sides.

在某些四边形中，角的性质更加具体。例如，在平行四边形中，对角相等，邻角之和为180°，因为它们是平行边上的同旁内角。

Sum of interior angles of a quadrilateral = 360° | 四边形内角和 = 360°

7. Interior Angles of Polygons | 多边形内角

A polygon with n sides can be divided into (n – 2) triangles from one vertex. Therefore, the sum of the interior angles of an n-sided polygon is (n – 2) × 180°.

一个n边形可以从一个顶点分成(n – 2)个三角形。因此，n边形内角之和为(n – 2) × 180°。

For example, a pentagon has 5 sides, so its interior angles add up to (5 – 2) × 180° = 540°. A hexagon has 6 sides, giving (6 – 2) × 180° = 720°.

例如，五边形有5条边，所以其内角和为(5 – 2) × 180° = 540°。六边形有6条边，内角和为(6 – 2) × 180° = 720°。

You can use this formula to find the sum of interior angles for any polygon, whether it is regular or irregular. If the polygon is regular, divide the sum by n to find each interior angle.

你可以用这个公式求任意多边形的内角和，无论它是正多边形还是不规则多边形。如果多边形是正多边形，将总和除以n即可得到每个内角。

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

Each interior angle of a regular n-gon = (n − 2) × 180° ÷ n | 正n边形每个内角 = (n − 2) × 180° ÷ n

8. Exterior Angles of Polygons | 多边形外角

An exterior angle is formed by extending one side of a polygon at a vertex. The exterior angle and its adjacent interior angle always add up to 180° because they lie on a straight line.

外角是通过延长多边形顶点处的一条边形成的。外角与其相邻的内角之和始终为180°，因为它们位于同一直线上。

The sum of the exterior angles of any convex polygon is always 360°, no matter how many sides it has. This is because as you walk around the polygon, you complete one full turn.

任何凸多边形的外角之和始终为360°，无论它有多少条边。这是因为当你绕多边形走一圈时，你完成了一个完整的旋转。

For a regular polygon, all exterior angles are equal. Therefore, each exterior angle of a regular n-sided polygon is 360° ÷ n. This is often faster than using the interior angle formula.

对于正多边形，所有外角都相等。因此，正n边形每个外角为360° ÷ n。这通常比使用内角公式更快。

Sum of exterior angles = 360° | 外角和 = 360°

Each exterior angle of a regular n-gon = 360° ÷ n | 正n边形每个外角 = 360° ÷ n

9. Regular Polygons and Symmetry | 正多边形与对称性

A regular polygon has equal side lengths and equal angles. Its symmetry properties can help you solve angle problems without memorising every interior angle value.

正多边形具有相等的边长和相等的角。它的对称性可以帮助你解决角度问题，而无需记住每个内角的值。

For example, a regular octagon has 8 equal exterior angles, so each exterior angle is 360° ÷ 8 = 45°. The adjacent interior angle is 180° – 45° = 135°. This is very useful in tiling and tessellation questions.

例如，正八边形有8个相等的外角，所以每个外角为360° ÷ 8 = 45°。相邻的内角为180° – 45° = 135°。这在铺砖和镶嵌问题中非常有用。

You should be able to move between interior and exterior angles quickly. If you know the interior angle, the exterior angle is 180° – interior angle. If you know the exterior angle, the interior angle is 180° – exterior angle.

你应该能够快速在内角和外角之间转换。如果你知道内角，外角为180° – 内角。如果你知道外角，内角为180° – 外角。

Regular polygon \| 正多边形	Number of sides n \| 边数n	Exterior angle \| 外角	Interior angle \| 内角
Equilateral triangle \| 等边三角形	3	120°	60°
Square \| 正方形	4	90°	90°
Regular hexagon \| 正六边形	6	60°	120°
Regular octagon \| 正八边形	8	45°	135°

10. Multi-step Angle Reasoning and Proof | 多步骤角度推理与证明

Higher tier questions often ask you to find an unknown angle by combining several rules. You must show each step clearly and write the angle fact you are using next to your calculation.

高阶题目通常会要求你结合多个规则求出未知角。你必须清晰地展示每一步，并在计算旁边写出你使用的角事实。

A common approach is to start with the angle you know, use parallel line rules to find equal or supplementary angles, then apply triangle or polygon angle sums. State reasons such as ‘angles on a straight line’ or ‘corresponding angles are equal’.

一种常见方法是从已知角出发，利用平行线规则找到相等或互补的角，然后应用三角形或多边形内角和。说明理由，例如’直线上的角’或’同位角相等’。

Proof questions may ask you to show that two angles are equal or that lines are parallel. To do this, identify the angle relationships and use algebra to connect them logically.

证明题可能会要求你证明两个角相等或两条直线平行。为此，识别角关系，并用代数将它们逻辑地联系起来。

Example: If one interior angle of a regular polygon is 150°, find the number of sides. The exterior angle is 180° – 150° = 30°. Since each exterior angle is 360° ÷ n, we have n = 360° ÷ 30° = 12.

例题：如果一个正多边形的一个内角为150°，求它的边数。外角为180° – 150° = 30°。因为每个外角为360° ÷ n，所以n = 360° ÷ 30° = 12。

11. Common Mistakes and Exam Tips | 常见错误与考试技巧

One common mistake is mixing up interior and exterior angles. Remember that an exterior angle is always supplementary to its adjacent interior angle, not equal to it.

一个常见错误是混淆内角和外角。请记住，外角与其相邻内角始终互补，而不是相等。

Another common error is applying the triangle angle sum of 180° to a quadrilateral. A quadrilateral has an interior angle sum of 360°, so always check the shape before adding angles.

另一个常见错误是将三角形内角和180°误用于四边形。四边形的内角和为360°，所以在相加之前一定要检查形状。

When working with parallel lines, be careful to identify whether the angle pair is equal or supplementary. Corresponding and alternate angles are equal, but co-interior angles add up to 180°.

在处理平行线时，要仔细辨别角对是相等还是互补。同位角和内错角相等，但同旁内角之和为180°。

Always give a reason for each angle fact you use. Exam mark schemes often award method marks for stating the correct reason, even if your final answer is incorrect.

对你使用的每个角事实都要给出理由。考试评分方案通常会给陈述正确理由的方法分，即使你的最终答案不正确。

12. Worked Practice Questions | 练习题解析

Try this question before reading the solution: In a regular polygon, each exterior angle is 24°. Work out the number of sides and the size of each interior angle.

在阅读解析之前先尝试这道题：在一个正多边形中，每个外角为24°。求出边数和每个内角的大小。

Since the exterior angles sum to 360°, the number of sides is n = 360° ÷ 24° = 15. Each interior angle is 180° – 24° = 156°.

因为外角之和为360°，边数为n = 360° ÷ 24° = 15。每个内角为180° – 24° = 156°。

Now try a parallel line question: AB is parallel to CD. A transversal crosses both lines. If one alternate angle is 72°, find the other alternate angle and the co-interior angle on the same side.

再试一道平行线题：AB平行于CD。一条截线穿过这两条直线。如果一个内错角为72°，求另一个内错角和同侧的同旁内角。

The other alternate angle is also 72° because alternate angles are equal. The co-interior angle is 180° – 72° = 108° because co-interior angles add up to 180°.

另一个内错角也是72°，因为内错角相等。同旁内角为180° – 72° = 108°，因为同旁内角之和为180°。

Finally, a multi-step problem: One angle in a triangle is 40°. The triangle is isosceles. Find the two possible sizes of the other angles.

最后是一道多步骤题：三角形中一个角为40°。该三角形是等腰三角形。求另外两个角的两种可能大小。

Case 1: The 40° angle is the unequal angle. The other two equal angles are (180° – 40°) ÷ 2 = 70° each. Case 2: The 40° angle is one of the equal angles, so the other equal angle is also 40°, and the third angle is 180° – 40° – 40° = 100°.

情况1：40°角是不相等的角。另外两个相等角为(180° – 40°) ÷ 2 = 70°。情况2：40°角是其中一个相等角，所以另一个相等角也是40°，第三个角为180° – 40° – 40° = 100°。

Published by TutorHao | Maths Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Unit 06 Angles: Parallel Lines, Polygons and Reasoning | 第六单元 角度：平行线、多边形与推理