Cambridge KS3 Maths: Solving Angle Equations | 剑桥 KS3 数学：求解角度方程

📚 Cambridge KS3 Maths: Solving Angle Equations | 剑桥 KS3 数学：求解角度方程

In many Cambridge KS3 mathematics exercises, especially questions like the one on page 211, students meet problems that connect geometry with algebra. A typical example gives the angles of a triangle as expressions in x and asks you to find x by using the angle sum. This article shows a clear method for these questions, including setting up the equation, solving it, and checking your answer.

在剑桥 KS3 数学的许多练习中，尤其是类似第211页第2题的问题，学生会遇到几何与代数结合的题目。一个典型例子是给出三角形内角的 x 表达式，要求你利用内角和求出 x。本文展示这类题目的清晰解法，包括建立方程、求解和检验答案。

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1. The Core Question | 核心问题

A common question from Cambridge Checkpoint practice is: 'The angles of a triangle are x°, (2x + 10)° and (3x - 10)°. Find the value of x and write down the size of each angle.'

剑桥 Checkpoint 练习中常见的一道题是："三角形的三个内角分别为 x°、(2x + 10)° 和 (3x - 10)°。求 x 的值，并写出每个角的大小。"

This type of question appears again and again in KS3 progression tests because it tests two skills at once: knowing the angle sum of a triangle and solving a linear equation.

这类题目在 KS3 进阶测试中反复出现，因为它同时考查两项技能：掌握三角形内角和以及解一元一次方程。

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2. Key Angle Rule: Angles in a Triangle | 关键角度规则：三角形内角和

The most important fact is that the interior angles of a triangle always add up to 180°.

最重要的事实是：三角形的三个内角之和始终为 180°。

Angle A + Angle B + Angle C = 180°

This is true for every triangle, whether it is right-angled, isosceles or scalene. If you cut out the three corners of any triangle and place them side by side, they form a straight line, which measures 180°.

这对任何三角形都成立，无论是直角三角形、等腰三角形还是不等边三角形。如果你把任意三角形的三个角剪下来并拼在一起，它们会形成一条直线，而直线正好是 180°。

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3. Turning Angle Language into Algebra | 把角度语言转化为代数式

Each angle is already given as an algebraic expression: x, 2x + 10 and 3x - 10. The variable x stands for an unknown number of degrees.

每个角已经用代数式给出：x、2x + 10 和 3x - 10。变量 x 代表未知的度数。

Do not let the addition or subtraction worry you: 2x + 10 means 'two times x, then add 10'. Likewise, 3x - 10 means 'three times x, then subtract 10'.

不要被加减号困扰：2x + 10 表示 "x 的两倍再加 10"。同样，3x - 10 表示 "x 的三倍再减 10"。

You can label the expressions clearly before solving:

在解题前，你可以先清楚地标出各表达式：

• x° (the smallest angle expression / 最小角表达式)
• (2x + 10)° (the middle angle expression / 中间角表达式)
• (3x - 10)° (the largest angle expression / 最大角表达式)

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4. Setting Up the Equation | 建立方程

Add the three angle expressions and set the sum equal to 180°.

把三个角度表达式相加，并让和等于 180°。

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

The brackets are not essential here, but they help you see the three separate angle terms. Writing the equation clearly is also important for earning method marks in a test.

这里的括号不是必须的，但它们能帮助你看出三个独立的角度项。清晰地写出方程对在考试中获得方法分也很重要。

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5. Solving the Equation Step by Step | 逐步解方程

First collect like terms on the left-hand side: x + 2x + 3x = 6x, and 10 - 10 = 0.

首先在左边合并同类项：x + 2x + 3x = 6x，同时 10 - 10 = 0。

6x = 180

Now divide both sides by 6 to isolate x.

现在把两边同时除以 6，以求出 x。

x = 180 ÷ 6 = 30

So the value of x is 30. Always show this division step, even if you can do it mentally.

因此 x 的值是 30。即使你能心算，也请一定展示这个除法步骤。

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6. Checking the Angles | 回代检验角度

Substitute x = 30 back into each expression:

把 x = 30 代回每个表达式：

• x° = 30°
• 2x + 10 = 2(30) + 10 = 70°
• 3x - 10 = 3(30) - 10 = 80°

You can also organise the checking work in a table:

你还可以用表格来整理检验过程：

Expression / 表达式	Value / 角度值
x	30°
2x + 10	70°
3x - 10	80°

Check the sum: 30 + 70 + 80 = 180°, so the answer is correct.

检查总和：30 + 70 + 80 = 180°，因此答案正确。

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7. Common Mistakes to Avoid | 需要避免的常见错误

Many students forget that the sum must be 180°, or they add the numbers before collecting x terms incorrectly.

许多学生忘记内角和必须是 180°，或者在合并 x 项之前错误地处理数字。

ommon error is writing 2x + 10° as 2(x + 10). These are not the same: 2(x + 10) = 2x + 20.

另一个常见错误是把 2x + 10° 写成 2(x + 10)。它们并不相同：2(x + 10) = 2x + 20。

Also, do not drop the degree symbol halfway through the working. Keep using ° with each angle until the final answer.

此外，不要在解题过程中丢掉度数符号。每个角的表达式中要保留 °，直到最终答案。

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8. Extending to Quadrilaterals and Polygons | 拓展到四边形和多边形

The same method works for quadrilaterals, but the interior angle sum is 360°.

同样的方法也适用于四边形，但四边形的内角和是 360°。

Angle A + Angle B + Angle C + Angle D = 360°

For example, if the angles of a quadrilateral are x, 2x, 3x and 4x, then:

例如，如果一个四边形的四个角分别是 x、2x、3x 和 4x，那么：

x + 2x + 3x + 4x = 360

10x = 360

x = 36

This pattern continues for higher polygons, using the general formula for the interior angle sum.

这个规律可以推广到更多边形，使用内角和的一般公式：

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

Here n is the number of sides

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