🏠 首页 CHAPTER 6 · ANGLES AND PARALLEL LINES 6.3

6.3

Parallel Lines and Transversals

💡 平行线永远不相交。当第三条线（截线 / transversal）穿过两条平行线时，会形成 8 个角。这些角中有 3 种特殊关系：
• 同位角（corresponding angles）—— 相等
• 内错角（alternate angles）—— 相等
• 同旁内角（co-interior / interior angles）—— 互补（和为 180°）

AParallel Lines

When two straight lines on the same plane do not intersect, they are called parallel lines. In the diagram below, $AB$ and $CD$ are parallel lines. We say "$AB$ is parallel to $CD$" and denote this by  $AB \parallel CD$.

C D A B

Let us look at some examples of lines which are not parallel to each other.

$EF$ and $GH$ are non-parallel lines as they intersect at point $O$.

O E H G F

$RS$ and $PQ$ are non-parallel lines as $RS$ produced and $PQ$ produced meet at point $T$ (that is, when $RS$ and $PQ$ are extended, they meet at $T$).

R P S Q T

Parallel lines are indicated by an equal number of arrowheads on lines parallel to each other pointing in one direction. Different sets of parallel lines are marked with different numbers of arrowheads.

B D A C One pair · AB ∥ CD E G F H Two pairs · EF ∥ HG, EH ∥ FG Three pairs of parallel lines

BTransversals

When a line intersects two other lines, the line is called a transversal of those two lines. In the diagram, $TS$ is a transversal of the lines $AB$ and $CD$. There are 4 angles formed at each point of intersection.

C D A B T S q p r s b a c d transversal

Let us look at how we can identify these pairs of angles according to their relative positions.

(a) The following are pairs of corresponding angles:

(i) ∠a and ∠p
p a
(ii) ∠b and ∠q
q b
(iii) ∠c and ∠r
r c

(b) The following are pairs of alternate angles:

(i) ∠a and ∠r
r a
(ii) ∠b and ∠s
s b

(交错于截线两侧，分布在两条平行线之间。)

(c) The following are pairs of interior angles on the same side of the transversal (co-interior angles):

(i) ∠a and ∠s
s a
(ii) ∠b and ∠r
r b

Suppose the lines $AB$ and $CD$ are parallel as shown in the diagram. Then the pairs of corresponding angles, alternate angles and interior angles will have interesting properties.

🧪 Activity 1 · Properties of angles between parallel lines

Objective: To investigate properties of angles formed by two parallel lines and a transversal.

$AB$ is parallel to $CD$. A transversal $ST$ cuts $AB$ and $CD$ at $Y$ and $Z$ respectively. From experiment (using ruler and protractor, or dynamic geometry software):

C D A B T S Z Y p = 50° r = 50° s = 130° a = 50°

From Activity 1, we observe the following properties:

When two parallel lines are cut by a transversal,
  • their corresponding angles are equal,  e.g., $\angle a = \angle p$  corr. ∠s, AB ∥ CD
  • their alternate angles are equal,  e.g., $\angle a = \angle r$  alt. ∠s, AB ∥ CD
  • the sum of their interior angles on the same side of the transversal is 180° (i.e., supplementary),  e.g., $\angle a + \angle s = 180°$  int. ∠s, AB ∥ CD

📌 SPOTLIGHT (Converse): Conversely, when two lines $AB$ and $CD$ are cut by a transversal and if
• their corresponding angles are equal, or
• their alternate angles are equal, or
• the sum of their interior angles is 180° (i.e., supplementary),
then $AB$ is parallel to $CD$.

📖 Worked Example 5

In the diagram, $AB \parallel CD$.

E G F H A B C D 78° 72° a° b° c°
  1. (a) Find the values of $a$, $b$ and $c$.
  2. (b) Determine whether $EF$ is parallel to $GH$. Explain your answer.

SOLUTION

(a)
$a = 78$  (corr. ∠s, AB ∥ CD)
$b + 72 = 180$  (int. ∠s, AB ∥ CD)
$b = 180 - 72 = \mathbf{108}$
∴ $b = 108$
$c = 72$  (alt. ∠s, AB ∥ CD)

(b) $a + b = 78 + 108 = 186 \neq 180$. The interior angles do not add up to 180°. ∴ $EF$ is not parallel to $GH$.

In Worked Example 5, suggest other methods to prove that $EF$ is not parallel to $GH$.
💡 思考

另一些验证方式：
• 检查 $EF$ 上 $78°$ 的同位角应等于 $GH$ 上的同位角。$GH$ 在下方的同位角 = ? 如果不等于 $78°$ 就不平行。
• 用内错角：$EF, GH$ 中如果交错角不等也证不平行。
关键是要找到一对角不满足"对应/交错/同旁内角"的关系。

✍️ Try It Yourself 5

In the diagram, $EF \parallel GH$. Given angles $58°$ on $EF$, $120°$ at $G$ on $GH$, $a°$, $b°$, $c°$ on transversals $AB$ and $CD$.

  1. (a) $a = $ $\;$ $b = $ $\;$ $c = $ 检查
    💡 思路

    $a$ = 58° (alt./corr. ∠s, EF ∥ GH); $b$ = 120° (corr.); $c$ = 180 - 120 = 60° (int. ∠s).

  2. (b) Determine whether $AB$ is parallel to $CD$. Explain.
    💡 思路

    检查同位角 / 内错角 / 同旁内角的关系。如果给定数据满足任一条性质，则平行；不满足则不平行。

📖 Worked Example 6 · Two pairs of parallel lines

In the diagram, $CDE$ and $ADG$ are straight lines with $AB \parallel CE$ and $AG \parallel CF$. Given angle 54° at $C$ (between $CE$ and $CF$), find the values of $x$ and $y$ where $x°$ is at $D$ and $y°$ is at $D$ (between $AB$-direction and $A$-side).

A B E G F C D 54° x° y°

SOLUTION

  • $\angle CDG + 54° = 180°$  (int. ∠s, CF ∥ AG)
  • $x + x + 54 = 180$  (if the two angles labelled $x°$ are equal)
  • $2x = 126$
  • $x = \mathbf{63}$
  • $\angle EDG = 54°$  (corr. ∠s, CF ∥ AG)
  • $y = \angle EDG$  (corr. ∠s, AB ∥ CE)
  • ∴ $y = \mathbf{54}$

📌 NOTE: To find $y°$, we need to find $\angle EDG$ or $\angle ADC$ first.

✍️ Try It Yourself 6

In the diagram, $BHC$ and $DHE$ are straight lines with $AB \parallel DE$ and $BC \parallel EF$. $\angle ABH = 72°$, $\angle GHC = 23°$. Find the values of $x$ and $y$ ($x°$ between $BHC$ direction and $E$, $y°$ between $BH$ and $HG$).

$x = $ ; $y = $ 检查

💡 思路

$AB \parallel DE$ 以及 $BC \parallel EF$ 给出平行四边形的角对应关系。
$x = \angle ABH = 72°$ (alt. or corr.).
$y = 180 - (72 + 23) = 85°$ (st. line through $H$).

📖 Worked Example 7 · Auxiliary line construction

In the diagram, $BA \parallel DE$. $\angle ABC = 30°$ and $\angle CDE = 35°$. Find $\angle x = \angle BCD$.

B A D E C F 30° 35° x x₁ x₂

ANALYSIS

$BA \parallel DE$ and the unknown angle $x$ is at $C$. No relationship can be seen directly. Hence we draw a line passing through point $C$ and parallel to $BA$ and $DE$ to establish a relationship between $\angle x$ and the parallel lines. Then $\angle x$ is made up of two smaller angles, $\angle x_1$ and $\angle x_2$. We can apply the properties we have learnt to find $\angle x_1$ and $\angle x_2$.

SOLUTION

Construct the line $FC$ parallel to the line $BA$.

  • $\angle x_1 = 35°$  (alt. ∠s, FC ∥ DE)
  • $\angle x_2 = 30°$  (alt. ∠s, FC ∥ BA)
  • $\angle x = \angle x_1 + \angle x_2 = 35° + 30° = \mathbf{65°}$

📌 NOTE: Construct the line $FC$ as a dotted line.

✍️ Try It Yourself 7
  1. (a) In the diagram, $BA \parallel DE$. $\angle ABC = 55°$ and $\angle CED = 20°$. Find $\angle x = \angle BCE$ (at vertex $C$ of the zigzag).
    $x = $ ° 检查
    💡 思路

    Construct line through $C$ parallel to $BA$ (and $DE$). $x = 55 + 20 = 75°$.

  2. (b) In the diagram, $AB \parallel FE$. $\angle ABC = 70°$, $\angle CDE = 120°$, reflex angle at point $D$ on the same side as $C$ is $250°$. Find $\angle a = \angle DEF$ (at $E$).
    $a = $ ° 检查
    💡 思路

    Construct one or more lines through $C, D$ parallel to $AB$ (and $FE$). The folded path gives: $a = 70 + (360 - 250) - 120 = 70 + 110 - 120 = 60°$. (Exact path depends on diagram; multiple constructions converge.)

PPractice Exercise 6.3

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

BASIC MASTERY
  1. Q1. Find the unknown marked angles in each diagram.
    (a) $AB \parallel CD$, transversal cuts at $G$ with $\angle EFD = 50°$. Find $a, b, c$ at $G$.
    $a = $ ; $b = $ ; $c = $ 检查
    💡 思路

    $a = 50$ (corr.); $b = 50$ (vert opp $a$); $c = 180 - 50 = 130$ (linear pair).


    (b) $AB \parallel CD$ cut by transversal; $\angle EBC = 113°$. Find $f, g$.
    $f = $ ; $g = $ 检查
    💡 思路

    $f = 113°$ (corr. or vert opp); $g = 67°$ (supp.).


    (c) Two parallel lines cut by transversal; given $110°$ and $24°$. Find $h$ and $k$.
    $h = $ ; $k = $ 检查
    💡 思路

    取决于位置：可能 $h = 180 - 110 = 70$ (co-interior); $k = 180 - 24 = 156$ (co-interior) 或 $k = 24$ (corr.).


    (d) Triangular setup with parallel sides and given $47°$ and $62°$. Find $p, q, r$.
    $p = $ ; $q = $ ; $r = $ 检查
    💡 思路

    $p$ alt 62°，$q$ alt 47°，$r = 180 - p - q$ 在三角形内（如果是三角形）= $180 - 62 - 47 = 71°$.

  2. Q2. Find the value of $x$ in each diagram.
    (a) $AB \parallel FC$, $\angle EBA = 75°$, find $x$ at $C$. → 检查
    💡 思路

    取决于位置：corr. ∠s → x = 75°; 或 co-int. → x = 105°.


    (b) $CD \parallel AB$ (with arrowmarks), 64° given, $x$ at $S$. → 检查
    💡 思路

    同位角或对顶角：$x = 64°$（或 supp. $116°$ 取决于具体角）。


    (c) Zigzag with parallel arrows: $35°$ at top, $68°$ at middle vertex, $x°$ at bottom. → 检查
    💡 思路

    过中间顶点作辅助平行线，z 形展开 → $x = 35 + 68 = 103°$.


    (d) $AB \parallel CDEF$, $113°$ given. → $x = $ 检查
    💡 思路

    同旁内角 $x + 113 = 180 \Rightarrow x = 67°$. 或 alt. → $x = 113°$.

INTERMEDIATE
  1. Q3. Find the value of $y$ in each diagram (involves auxiliary line construction).
    (a) Z-shape: 153° and 115° given; $y°$ at apex. → 检查
    💡 思路

    过顶点作辅助平行线把 $y$ 分成 $y_1 + y_2$，再用 alt. ∠s 和 supp.。


    (b) Zigzag with $20°$, $65°$, $30°$ given; $y°$ at vertex.
    $y = $ 检查
    💡 思路

    过顶点构辅助平行线。把 $y$ 拆成两个角加和。具体值取决于图示。

  2. Q4. $AC \parallel DE$ and $BCD$ is a straight line. $\angle ACF = 35°$ and $\angle BDE = 110°$. Find $x$ and $y$ (positions at $C$ and $D$).
    $x = $ ; $y = $ 检查
    💡 思路

    Use alt./corr. angles with $AC \parallel DE$. Specific values from diagram.

  3. Q5. In the diagram, $AB$ and $CD$ are cut by the transversals $EF$ and $EG$. Given $\angle EAB = 75°$, $\angle ECF = 120°$, find $\angle a = \angle DCG$.
    (a) Is $AB$ parallel to $CD$? 检查
    (b) $a = $ ° 检查
    💡 思路

    检查同位角或内错角是否相等。具体取决于角的位置。

ADVANCED
  1. Q6. The diagram represents part of a flight of steps where $BA \parallel CD \parallel EF$, $BC \parallel DE$ and $\angle ABC = 105°$. Find the angles $x = \angle BCD$ and $y = \angle DEF$.
    $x = $ ° ; $y = $ ° 检查
    💡 思路

    "台阶"图案：每步交错。$BA \parallel CD$ 且 $BC$ 是截线 → $x + 105 = 180 \Rightarrow x = 75°$（co-int.）。同理对 $DE$/$EF$：$y$ 与 $\angle ABC$ 对应 → $y = 105°$.

  2. Q7. The diagram shows a side view of a house, where $ABCDE$ is its roof and $\angle ACE = 106°$ at the apex $C$. Given that $GB \parallel FD$, find $\angle x$ (an angle in the diagram).
    $x = $ ° 检查
    💡 思路

    取决于 $x$ 是哪个角。屋顶常假设对称：左斜面和右斜面相等。利用平行线找到 $x$ 与 $106°$ 的关系。

  3. Q8. In the diagram, $BA \parallel DE$ forming a zigzag with marked angles $a, e$ on outside; $b, c, d$ on a zigzag path. Form an equation connecting
    (a) $a$ and $e$. → 检查
    (b) $b$, $c$ and $d$. → 检查
    💡 思路

    (a) $a$ 和 $e$ 都是 $BA, DE$ 平行线上的同位/交错角 → $a = e$.
    (b) 经典 "锯齿" 关系：中间顶点 $C$ 的转角 $c = b + d$（左右两边平行线展开累加）。

  4. Q9. The diagram (not drawn to scale) shows a zigzag where $140°$ is given at top, $x°$ in the middle, $30°$ in the next zigzag, $y°$ at the bottom.
    (a) Express angle $x$ in terms of angle $y$. → $x = $ 检查
    (b) Given that angle $x$ is acute, explain why angle $y$ is acute.
    💡 思路

    (a) 用辅助线展开 zigzag：$x + y = 140 + 30 = 170 \Rightarrow x = 170 - y$.
    (b) $x$ 锐角 → $x < 90$ → $170 - y < 90$ → $y > 80$. $y > 80$ 但还需 $y < 90$ 才锐角 → 进一步分析 $y$ 取值范围。如果题目本意是 $x + y = 170$ 且 $x$ 锐角 → $y$ 必须 $> 80°$。要让 $y$ 也是锐角必须 $80° < y < 90°$，所以 $y$ 是锐角范围内的窄区间。

  5. Q10. Three lines $l$, $m$, $n$ are in the same plane. When $l$ is perpendicular to both $m$ and $n$, can we conclude that $m$ is parallel to $n$? Explain.
    Answer: 检查
    💡 思路

    是。$l \perp m$ 意味着 $l$ 和 $m$ 的夹角是 90°；$l \perp n$ 意味着 $l$ 和 $n$ 的夹角也是 90°。所以 $m$ 和 $n$ 与同一截线 $l$ 形成相等的同位角 → $m \parallel n$（同位角相等 ⇒ 两线平行的逆命题）。

  6. Q11. A rectangle $ABCD$ is cut along $EF$ as shown. The piece $EFCD$ is flipped so that the two pieces form an L-shaped figure $ABEDCF$ and $\angle AFC = 90°$. In the L-shape, find
    (a) $\angle AFE$. → ° 检查
    (b) $\angle BED$. → ° 检查
    💡 思路

    翻折后 $EF$ 重合自身但方向反转。具体角度依图示，通常 $\angle AFE = 45°$（折痕和直角的二等分），$\angle BED = 135°$ 等。