🏠 首页 CHAPTER 6 · 章末 SUM UP

📋 Let's Sum Up!

Points, Lines and Planes

plane ACDE line AB line segment AE ray AC point A

Classification of Angles

According to sizes

a (a) ∠a<90° b (b) ∠b=90° c (c) 90°<∠c<180° d (d) 180°<∠d<360°

According to the relationship between angles

  • (a) If the sum of two angles is 90°, then the two angles are called complementary angles.
  • (b) If the sum of two angles is 180°, then the two angles are called supplementary angles.
  • (c) If two angles share a common side and a common vertex but do not overlap, then the two angles are called adjacent angles.

Properties of Angles

Adjacent Angles on a Straight Line

a b c

$\angle a + \angle b + \angle c = 180°$

(adj. ∠s on a st. line)

Angles at a Point

a b c d

$\angle a + \angle b + \angle c + \angle d = 360°$

(∠s at a point)

Vertically Opposite Angles

a b c d

$\angle a = \angle d$, $\angle b = \angle c$

(vert. opp. ∠s)

Parallel Lines and Transversals

Two parallel lines $AB$ and $CD$ are cut by a transversal $TS$.

C D A B T S q p r s b a c d
  • $\angle a = \angle p$  (corr. ∠s, AB ∥ CD)
  • $\angle a = \angle r$  (alt. ∠s, AB ∥ CD)
  • $\angle a + \angle s = 180°$  (int. ∠s, AB ∥ CD)
📝 Review Exercise 6

State the angle properties which you have applied in your working.

  1. Q1. Find the value of $x$ in each diagram.
    (a) $ABC$ is a straight line with angles $2x°$, $(4x-3)°$, $63°$ at $B$. → $x = $ 检查
    (b) At a point: $3x°$, $45°$, $4x°$, $2x°$ around it. → $x = $ 检查
    💡 思路

    (a) $2x + (4x-3) + 63 = 180 \Rightarrow 6x + 60 = 180 \Rightarrow x = 20$.
    (b) $3x + 45 + 4x + 2x = 360 \Rightarrow 9x = 315 \Rightarrow x = 35$.

  2. Q2. Find the values of $x$ and $y$.
    (a) $AOB$ and $COD$ are straight lines intersecting at $O$; angles labelled $y°$, $y°/4$, $4x°$.
    $y = $ ; $x = $ 检查
    💡 思路

    $y$ 与 $4x$ 是对顶角 → $4x = y$. $y + y/4 = 180$ (adj on st line) → $\tfrac{5y}{4} = 180 \Rightarrow y = 144$, $x = 36$.


    (b) $ABE$ is a straight line; parallelogram-like diagram with $2x°$, $x°$, $y°$ at appropriate vertices.
    $x = $ ; $y = $ 检查
    💡 思路

    用对应/邻补角关系。具体值取决于图示，常见答案 $x = 60°, y = 120°$。

  3. Q3. A squash ball hits a vertical wall $HK$ along path $AB$ and rebounds along $BC$. Given $\angle ABC = 56°$. Find $y$.
    $y = $ ° 检查
    💡 思路

    反射角 = 入射角；两角分别为 $y°$，加 $\angle ABC = 56°$ 合 180° (st line) → $2y = 124 \Rightarrow y = 62°$。

  4. Q4. A piece of wire is bent into a parallelogram shape with $BA \parallel CE$, $BC \parallel AE$. If $\angle ABC = 110°$, find $\angle x = \angle BCE$ and $\angle y = \angle AED$.
    $x = $ ° ; $y = $ ° 检查
    💡 思路

    平行四边形：相邻角和 = 180°，对角相等。$x = 180 - 110 = 70°$; $y$ 与 $\angle ABC$ 对角 → $y = 110°$.

  5. Q5. Find $\angle x$ in each zigzag diagram.
    (a) Three parallel lines with arrows; angles $75°$ at $B$, $x°$ at $D$. → $x = $ ° 检查
    (b) Zigzag: $110°$, $45°$ given; $x$ at vertex $C$. → $x = $ ° 检查
    (c) Zigzag: $100°$, $115°$ given; $x$ at vertex $E$. → $x = $ ° 检查
    💡 思路

    (a) alt./corr.: $x = 75°$. (b)(c) 构造辅助平行线展开。

  6. Q6. Flight of steps: $BA \parallel CD \parallel EF$, $BC \parallel DE$, $\angle ABC = 105°$. Find $x = \angle BCD$ and $y = \angle DEF$.
    $x = $ ° ; $y = $ ° 检查
  7. Q7. Side view of a house: $ABCDE$ is the roof, $\angle ACE = 106°$ (at the apex). Given $GB \parallel FD$, find $x$ (at some position in the diagram).
    $x = $ ° 检查
    💡 思路

    具体取决于 $x$ 在图中的位置。对称屋顶通常 $x = (180-106)/2 = 37°$ 或 $x = 180-106 = 74°$。

  8. Q8. In the diagram, $BA \parallel DE$. Form an equation connecting
    (a) $a$ and $e$ → 检查
    (b) $b, c, d$ → 检查
    💡 思路

    (a) $a = e$ (corr./alt.) 或 $a + e = 180$ (co-int.) 视位置而定。
    (b) Zigzag 经典结果：中间顶点处的角 = 两侧角之和，即 $c = b + d$。

  9. Q9. The diagram is not drawn to scale. Given $140°$ at top and $30°$ at lower part. Express $x$ in terms of $y$.
    $x = $ 检查
    💡 思路

    $x + y = 140 + 30 = 170 \Rightarrow x = 170 - y$.

  10. Q10. Three lines $l$, $m$, $n$ in same plane. When $l \perp m$ and $l \perp n$, can we conclude $m \parallel n$? Explain.
    Answer: 检查
    💡 思路

    $l \perp m, l \perp n \Rightarrow$ $l$ 与 $m, n$ 都成 90° 角 $\Rightarrow$ 同位角相等 $\Rightarrow m \parallel n$.

  11. Q11. A rectangle $ABCD$ is cut along $EF$ as shown. Piece $EFCD$ is flipped so the two pieces form an L-shaped figure $ABEDCF$ and $\angle AFC = 90°$. Find
    (a) $\angle AFE$ → ° 检查
    (b) $\angle BED$ → ° 检查
  12. Q12. Find $\angle p$ and $\angle q$ in the diagram. (Reflex angle 270° at $C$ in the middle, 42° at $A$.)
    $p = $ ° ; $q = $ ° 检查
    💡 思路

    反射角 270° → 非反射角 = 360 - 270 = 90°. 其他角由直线或对顶推导。

  13. Q13. In the diagram, $ACE$, $BCF$ and $DCG$ are straight lines and $AB \parallel HC$. Find $p, q, r, s$. (Given $\angle FCH = 20°$, and triangle base angles $50°, 65°$.)
    $p = $ ; $q = $ ; $r = $ ; $s = $ 检查
    💡 思路

    Use $AB \parallel HC$ → corr./alt. angles match base angles 50° and 65°. Then apply st line / vert opp.

  14. Q14. In the diagram, $AB \parallel CD \parallel EF$ and $\angle DCF = 35°$. Find the values of $a$ and $b$. ($a + 5b$ given somewhere.)
    $a = $ ; $b = $ 检查
    💡 思路

    Use the 3 parallel lines + transversals. Exact values depend on diagram positions.

  15. Q15. $\angle a$ and $\angle b$ are complementary ($a + b = 90$) and $\angle a$ and $\angle c$ are supplementary ($a + c = 180$). If $\angle a = x°$:
    (a) Express $\angle b$ in terms of $x$: .
         Express $\angle c$ in terms of $x$: . 检查
    (b) Find $\angle c - \angle b$. → ° 检查
    💡 思路

    $b = 90 - x$; $c = 180 - x$; $c - b = (180-x) - (90-x) = 90$.

📔 Maths Journal

✍️ 写下你的反思

1There are differences between corresponding angles and alternate angles between parallel lines. How would you identify these differences?

2Describe how you would verify if two lines are parallel to each other.

💡 参考

1. 区分同位角 / 内错角：
• 同位角（corresponding）：在截线同一侧，两条线同一侧（比如：都在左上方）。形成"F"字形。
• 内错角（alternate）：在截线两侧，两条线之间（"夹层"区域）。形成"Z"字形。
• 同旁内角（co-interior）：在截线同一侧，两条线之间。形成"C"或"U"字形。

2. 如何验证两线平行：
作一条截线穿过两线，测量任意一对：
• 同位角相等 → 平行 ✓
• 内错角相等 → 平行 ✓
• 同旁内角和为 180° → 平行 ✓
任何一个满足就能证明平行（互为逆定理）。如果都不满足，则不平行。