Solving Angles with Algebra: A Problem from p248 | 用代数解角度问题：来自p248的一题

📚 Solving Angles with Algebra: A Problem from p248 | 用代数解角度问题：来自p248的一题

This article unpacks a classic geometry problem taken from page 248 of the Cambridge KS3 Mathematics textbook. The question involves two parallel lines cut by a transversal, with angles expressed as algebraic expressions. You are required to form and solve a linear equation to find the unknown variable. Mastering this skill bridges algebra and geometry, sharpening logical reasoning and equation-solving techniques.

本文详细解析剑桥KS3数学教材第248页的一道经典几何题。题目给出两条平行线被一条截线所切，某些角用含有未知数的代数式表示。你需要建立并求解线性方程，找出该未知数。掌握这项技能能够连接代数与几何，锻炼逻辑推理和方程求解能力。

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1. Understanding the Problem | 理解问题

The diagram on page 248 typically shows two parallel lines intersected by a third line (the transversal). One angle is labelled (3x – 20)° and another angle, which is a corresponding angle relative to the first, is labelled (2x + 15)°. The task is to calculate the value of x and hence the actual angle measures. Recognising which angle pairs are related is the first critical step.

第248页的示意图通常展示两条平行线与第三条截线相交。其中一个角被标记为(3x – 20)°，另一个角是与它位置对应的同位角，标记为(2x + 15)°。题目要求计算x的值，并进一步求出角的实际度数。识别角与角之间的位置关系是关键的第一步。

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2. Angle Facts: Corresponding Angles | 角度事实：同位角

When two parallel lines are cut by a transversal, several pairs of angles are created. The most relevant ones here are corresponding angles. Corresponding angles sit on the same side of the transversal and in matching corners relative to the parallel lines. They are always equal in measure. This fact allows us to connect the two algebraic expressions.

当两条平行线被一条截线切割时，会形成多种角对关系。这里最相关的就是同位角。同位角位于截线的同一侧，并在两条平行线相对应的角落里。它们的度数始终相等。这一事实使我们能够将两个代数表达式联系起来。

Other angle pairs can also appear, such as alternate interior angles (equal) and co-interior angles (summing to 180°). Recognising them helps avoid confusion.

其他角对也可能出现，比如内错角（相等）和同旁内角（和为180°）。识别它们有助于避免混淆。

• Corresponding angles: equal
• Alternate interior angles: equal
• Co-interior (allied) angles: sum to 180°

• 同位角：相等
• 内错角：相等
• 同旁内角：和为180°

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3. Setting Up the Equation | 建立方程

Because the two marked angles are corresponding angles, their measures must be identical. We therefore write the equation:

由于这两个标记角是同位角，它们的度数必定相等。因此我们写出方程：

3x − 20 = 2x + 15

The expression on the left represents the first angle in degrees, and the expression on the right represents the second angle. The equation captures the condition that the lines are parallel; if they were not parallel, the angles would not necessarily be equal.

左边的表达式代表第一个角的度数，右边的表达式代表第二个角的度数。该方程体现了平行线的条件；如果直线不平行，这些角就不一定相等。

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4. Solving for x | 求解x

Solve the linear equation step by step. First, subtract 2x from both sides to collect the variable terms on one side.

逐步解此线性方程。首先，从两边同时减去2x，将含有变量的项集中到一侧。

3x − 20 − 2x = 2x + 15 − 2x

x − 20 = 15

Next, add 20 to both sides to isolate x.

接下来，两边同时加20，解出x。

x − 20 + 20 = 15 + 20

x = 35

The solution is x = 35. It is always wise to pause and check that your arithmetic is correct.

解为 x = 35。最好稍作停顿，检查计算是否正确。

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5. Verifying the Solution | 验证解

Substitute x = 35 back into both angle expressions to confirm they are indeed equal. The first angle becomes (3 × 35 − 20)° = (105 − 20)° = 85°. The second angle becomes (2 × 35 + 15)° = (70 + 15)° = 85°. Both give 85°, so the solution satisfies the original condition perfectly.

将x = 35代回两个角的表达式，确认它们确实相等。第一个角为 (3 × 35 − 20)° = (105 − 20)° = 85°。第二个角为 (2 × 35 + 15)° = (70 + 15)° = 85°。两者均为85°，因此解完全满足原题条件。

Verification is an essential habit, especially in exams, because it reveals any sign errors or misapplication of angle facts.

验证是一项重要习惯，尤其在考试中，因为它能揭示符号错误或对角的关系的错误应用。

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6. Finding the Angle Measures | 求出角的度数

Now that x is known, we can evaluate the size of each marked angle. Both corresponding angles are 85°. If the problem asked for the size of a vertically opposite angle or an alternate angle, you would apply the relevant angle fact to find it. For example, the vertically opposite angle to 85° is also 85°, and its supplementary angle is 180° − 85° = 95°.

知道x的值后，我们就可以求出每个标记角的大小。两个同位角均为85°。如果题目要求求对顶角或内错角的度数，只需应用相关角度关系即可。例如，85°的对顶角也是85°，而其补角为180° − 85° = 95°。

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7. Alternative Approach: Vertically Opposite Angles | 替代方法：对顶角

Sometimes the diagram places the given angles in different positions. Suppose the angle labelled (3x – 20)° and the angle labelled (2x + 15)° are not directly corresponding but are vertically opposite to corresponding angles. The reasoning still holds: vertically opposite angles are equal, and corresponding angles are equal, so the two given angles are still equal. Thus the equation remains unchanged. Understanding chains of equal angles builds confidence in more complex diagrams.

有时示意图把给定角放在不同位置。假设标记为(3x – 20)°的角和标记为(2x + 15)°的角不是直接的同位角，而是分别与同位角对顶。推理依然成立：对顶角相等，同位角相等，因此这两个给定角仍然相等。于是方程不变。理解角相等的传递关系能增强处理更复杂图形的信心。

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8. Common Pitfalls | 常见误区

One frequent mistake involves incorrect sign handling when moving terms. For example, a student may subtract 20 from 15 without adding it first, obtaining x = -5 instead of 35. Always perform the same operation on both sides.

一个常见错误是在移项时错误处理正负号。例如，学生可能从15中减去20而未先加20，得到x = -5而不是35。务必在等式两边执行相同运算。

Another pitfall is misidentifying the angle pair. If a student mistakes co-interior angles for corresponding angles, they might set the sum equal to 180°. That would lead to (3x – 20) + (2x + 15) = 180, giving x = 37, which is incorrect for the given diagram. Hence, double-check the angle relationship before writing the equation.

另一个误区是错误判断角的类型。如果学生将同旁内角误认为同位角，就可能设它们的和为180°。那会导致方程(3x – 20) + (2x + 15) = 180，解得x = 37，这对于原图来说是不正确的。因此，写方程前务必再次确认角之间的关系。

Forgetting to include the degree symbol or omitting parentheses can also lead to algebraic slips. Always write the equation clearly, and consider using brackets around each expression initially.

忘记写度数符号或漏掉括号也可能导致代数错误。务必清晰写出方程，并考虑一开始在每个表达式外加括号。

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9. Extending the Skill | 拓展技能

Once you are comfortable with corresponding angles, try variations. For example, if the problem gives an alternate interior angle relationship: (4x – 10)° and (3x + 25)° are alternate interior angles, set them equal and solve. The equation 4x – 10 = 3x + 25 gives x = 35 as well, but the angle measures become different. This shows that the equation structure is identical; only the constants change.

一旦你对同位角问题感到熟练，就可以尝试变式。例如，题目给出内错角关系：(4x – 10)°和(3x + 25)°是内错角，则设它们相等并求解。方程4x – 10 = 3x + 25同样解得x = 35，但角的度数会不同。这表明方程结构一致，只有常数变化。

Another common extension involves an algebraic angle and a constant angle that are supplementary. For instance, (2x + 30)° and 110° are co-interior angles. Set up 2x + 30 + 110 = 180, then solve to find x. This type of question tests your ability to switch between equality and sum-to-180 facts.

另一个常见拓展题涉及一个代数角与一个常数角互补。例如，(2x + 30)°和110°是同旁内角。列出方程2x + 30 + 110 = 180，然后求解x。这类题目考查你在相等关系与和为180°关系之间切换的能力。

Working through several p248-style problems solidifies the connection between parallel line geometry and linear equations, preparing you for more advanced topics like solving geometric problems using quadratic equations in later years.

多做几道类似于第248页的题目，能巩固平行线几何与线性方程之间的联系，为将来使用二次方程解决几何问题等更高级的主题做好准备。

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10. Conclusion | 总结

The p248 problem illustrates a fundamental KS3 skill: expressing geometric conditions as equations and solving them accurately. By identifying angle pairs, setting up simple linear equations, and verifying results, you build a reliable problem-solving routine. This approach works across the entire geometry syllabus, making it a valuable tool in your mathematical toolkit.

第248页的题目展示了KS3的一项基本技能：将几何条件表达为方程并准确求解。通过识别角的类型、建立简单线性方程并验证结果，你搭建起一套可靠的问题解决流程。这种方法适用于整个几何课程，是你数学工具箱中的宝贵武器。

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