6.2
Angles
AAngles and Measurements
When two line segments meet at a common point, they form an angle at that point. The point where the line segments meet is called a vertex. In the diagram, the angle with vertex $A$ and sides $AB$ and $AC$ are marked.
We denote this angle by
- $\angle BAC$ or $\angle CAB$ (an angle symbol and three capital letters)
- or $\angle x$. (an angle symbol and a small letter)
$\angle BAC$ can also be written as $B\hat{A}C$.
The commonly used unit for measuring an angle is the degree and it is denoted by the symbol °. A complete rotation about a point is equal to 360°.
BTypes of Angles
We classify an angle according to its size. Let us look at the different types of angles: acute angle, right angle, obtuse angle and reflex angle.
📌 SPOTLIGHT:
• A right angle is usually marked by the symbol "□" (small square in the corner).
• An angle of 180° is called a straight angle.
💡 思考
反射角范围：$180° < d < 360°$，所以 $d/2$ 范围：$90° < d/2 < 180°$。这是钝角的范围，不是锐角。所以"反射角的一半是锐角"不可能。
Look at the two diagrams below. Each diagram has two angles at points $B$ and $Q$. $\angle ABC$ is an acute angle and $\angle PQR$ is an obtuse angle. We refer to the reflex angles at points $B$ and $Q$ as reflex $\angle ABC$ and reflex $\angle PQR$, respectively.
Now, let us look at how we can classify different pairs of angles according to the relationship between each pair.
If the sum of two angles is 90°, then the two angles are called complementary angles.
Examples: $\angle ABD$ and $\angle DBC$ (both at $B$, summing to 90°); $\angle PQR$ and $\angle PRQ$ (in a right triangle).
If the sum of two angles is 180°, then the two angles are called supplementary angles.
If two angles share a common side and a common vertex but do not overlap, then the two angles are called adjacent angles.
📝 EXTRAINFO: Complementary angles and supplementary angles do not have to be adjacent but they can be.
📌 SPOTLIGHT: When two lines intersect at right angles, they are said to be perpendicular to each other. We write $AB \perp CD$. Two right angles form a pair of supplementary angles.
CProperties of Angles
Adjacent Angles on a Straight Line
In the diagram, $ABC$ is a straight line. The three angles, $\angle a$, $\angle b$ and $\angle c$, are adjacent angles on a straight line. $\angle a + \angle b + \angle c = 180°$.
(Abbreviation: adj. ∠s on a st. line)
In the diagram, $ABC$ is a straight line. Find the value of $x$.
SOLUTION
- $2x + 90 + x = 180$ (adj. ∠s on a st. line)
- $3x = 180 - 90$
- $3x = 90$
- $x = \dfrac{90}{3}$
- ∴ $x = \mathbf{30}$
📌 SPOTLIGHT: If an angle is marked with a letter and a degree symbol, such as $x°$, then $x$ represents a number and does not have a unit. The middle ray from $B$ in this diagram is perpendicular to $AC$ (right-angle mark not drawn but assumed for the 90° in the equation).
📌 NOTE: Always state the angle property which you have applied in your working.
- (a) In the diagram, $XYZ$ is a straight line. Find the value of $w$.
$w = $ 检查
💡 思路
三个角在 $XYZ$ 直线上：$w + 140 + w = 180 \Rightarrow 2w = 40 \Rightarrow w = 20$.
- (b) In the diagram, $AOB$ is a straight line. The angles at $O$ are $(a+55)°$ (between $OA$ and $OD$), $(35-a)°$ (between $OC$ and $OB$), and the rays $OC$, $OD$ make up the middle. What is the relationship between $OC$ and $OD$? Explain.
💡 思路
$AOB$ 直线，所以 $(a+55) + \angle COD + (35-a) = 180 \Rightarrow \angle COD = 180 - 90 = 90°$. 所以 $OC \perp OD$（互相垂直）。
Sum of Angles at a Point
In the diagram, $\angle w$, $\angle x$, $\angle y$ and $\angle z$ share a common vertex at $O$ and each angle is adjacent to two other angles. These four angles are called angles at a point.
$\angle w + \angle x + \angle y + \angle z = 360°$.
(Abbreviation: ∠s at a point)
📌 INVARIANCE: While all the angles on a straight line add up to 180°, all the angles at a point add up to 360°.
Find the value of $x$ in the diagram (four angles at a point: $3x°$, $x°$, $72°$, $4x°$).
SOLUTION
- $x + 3x + 4x + 72 = 360$ (∠s at a point)
- $8x + 72 = 360$
- $8x = 288$
- ∴ $x = \mathbf{36}$
Find the value of $x$ in the diagram (three angles at a point: $5x°$, $3x°$ on top, $x°$ on left).
$x = $ 检查
水平线已确定 180°；上方三个角加 5x、3x 都在直线上方，合 = $x + 5x + 3x = 9x = 180$？或者 4 个角合 360°？💡 思路
若三个角在直线上方 + 一个未标角在直线下方（180°）：$x + 5x + 3x = 180 \Rightarrow 9x = 180 \Rightarrow x = 20$。
若全部在一点周围：$x + 5x + 3x = 360 \Rightarrow 9x = 360 \Rightarrow x = 40$。
根据图示有"水平线"（直线在底）且三个角向上方分布，正确解读是直线上方：$9x = 180 \Rightarrow x = 20$.
Vertically Opposite Angles
In the diagram, two straight lines $AB$ and $CD$ intersect at the point $O$ and form four angles. The pairs of angles, $\angle a$ and $\angle b$ as well as $\angle x$ and $\angle y$, are called vertically opposite angles.
$\angle a = \angle b$ and $\angle x = \angle y$.
(Abbreviation: vert. opp. ∠s)
📌 SPOTLIGHT (Proof):
$\angle a + \angle x = 180°$ (adj. ∠s on a st. line)
$\angle b + \angle x = 180°$ (adj. ∠s on a st. line)
∴ $\angle a + \angle x = \angle b + \angle x \Rightarrow \angle a = \angle b$. ✓
💡 思考
"对顶角"必须是两条直线相交所形成的对位角。
• 如果两条线在同一交点相交且对位，✓
• 如果只有一条直线（另一条是折线/不通过交点）：✗
• 如果两条线不延伸过交点的另一侧：✗
In the diagram, $AD$, $BE$ and $CF$ are straight lines intersecting at $G$. The angle $\angle AGF = 80°$ and angles $y$ on both sides as shown. Find $\angle y$.
SOLUTION
- $\angle BGC = 80°$ (vert. opp. ∠s)
- $\angle AGB + \angle BGC + \angle CGD = 180°$ (adj. ∠s on a st. line)
- $\angle y + 80° + \angle y = 180°$
- $2\angle y = 100°$
- ∴ $\angle y = \mathbf{50°}$
📌 SPOTLIGHT: If an angle is marked without the symbol (°), such as $y$, then the final answer for $y$ must have a value and a unit which is degree.
In the diagram, $PS$, $QT$, $RU$ are straight lines intersecting at $V$. The marked angles are $3z$, $70°$, and $(2z + 15)°$. Find the value of $\angle z$.
$z = $ 检查
$P, V, S$ 共线，所以 $\angle PVU + \angle UVT + \angle TVS = 180°$，即 $3z + 70 + (2z+15) = 180$（如果 70 在 $UVT$ 位置）。💡 思路
$3z + 70 + 2z + 15 = 180 \Rightarrow 5z + 85 = 180 \Rightarrow 5z = 95 \Rightarrow z = 19$.
PPractice Exercise 6.2
- Q1. Classify each of the marked angles in the figure (a stylised dancer pose) as acute, obtuse, right or reflex.
(a) $\angle a$ → 检查
(b) $\angle b$ → 检查
(c) $\angle c$ → 检查
(d) $\angle d$ → 检查(分类基于角的大小；具体角值取决于课本图示。常见答案如上。)
- Q2. $AOB$ is a straight line. Find each unknown marked angle.
(a) $E$, $40°$, $a°$, $F$ marked at $O$ on st line $AOB$. The angles between $OE$ and $OA$ is given (40°), between $OE$ and $OF$ is $a°$, and on the other side of $OF$ also 40° (mirror). Find $a$. → 检查
(b) $P$, $R$, $B$, $Q$ angles with $61°, b°, 31°, 18°$ on st line $AOB$. → $b = $ 检查💡 思路
(a) $40 + a + 40 = 180 \Rightarrow a = 100°$.
(b) $61 + b + 31 + 18 = 180 \Rightarrow b = 70°$. - Q3. Find the unknown angle $x$ in each diagram.
(a) Four angles around a point: $64°$, $90°$ (right-angle mark), $x°$, $125°$. → $x = $ 检查
(b) Five angles around a point: $60°$, $x°$, $54°$, $78°$, $89°$. → $x = $ 检查💡 思路
(a) $360 - 64 - 90 - 125 = 81$.
(b) $360 - 60 - 54 - 78 - 89 = 79$. - Q4. Find the value of $y$ in the diagram (three equal angles $y°$, $y°$, $y°$ at a point).
$y = $ 检查💡 思路
$3y = 360 \Rightarrow y = 120°$.
- Q5. In each diagram, lines $AB$ and $CD$ intersect at a point. Find the unknown marked angles.
(a) $\angle EAC = 33°$ (or similar); find $x$ and $y$.
$x = $ $y = $ 检查💡 思路
$y$ 与 33° 是对顶角 → $y = 33°$. $x$ 与 33° 是邻角（一条直线上）→ $x = 180 - 33 = 147°$。
(b) Three lines through $F$; $76°, 128°$ given; find $p$ and $q$.
$p = $ $q = $ 检查💡 思路
两条直线交于 F，4 个角；$76°$ 和 $128°$ 是相邻的（合 = 204° ≠ 180°），所以实际上是三条直线。$p$ 与 76° 是对顶角 → $p = 76°$；$q$ 与 128° 是对顶角 → $q = 128°$。
- Q6. $AB$ is a plane mirror. A light ray $PQ$ hits the mirror at $Q$ and is reflected along $QR$ such that $\angle AQP = \angle BQR = x°$. If $\angle PQR = 110°$, find $x$.
$x = $ 检查💡 思路
$AQB$ 直线 → $x + 110 + x = 180 \Rightarrow 2x = 70 \Rightarrow x = 35°$。
- Q7. $LMN$ is a straight line. At $M$, an angle of 134° is given on the left (between $LM$ and a ray $MQ$ going down). Three rays $MP, MQ, MN$ on the right (from $M$): $y°$ (between $MN$ and one of the rays), $x°$ (between two rays).
(a) Find $y$. → 检查
(b) Type of angle $\angle PMN$? → 检查
Type of angle $\angle LMN$? → 检查💡 思路
(a) 134° 和 $y$ 在直线 $LMN$ 同一边相邻 → $y = 180 - 134 = 46°$.
(b) $\angle PMN$ < 90° → 锐角。
$\angle LMN = 180°$ → 平角 (straight angle). - Q8. Find the value of $x$ in the diagram (around a point: $3x°$, $50°$, $x°$, $4x°$, $70°$).
$x = $ 检查💡 思路
$3x + 50 + x + 4x + 70 = 360 \Rightarrow 8x + 120 = 360 \Rightarrow x = 30$.
- Q9. (OPEN) In the diagram, $E$, $F$, $D$ are rays from $B$ on st. line $ABC$. Name (a) two acute angles, (b) two obtuse angles in the diagram.
Two acute angles: 检查
Two obtuse angles: 检查(开放题，根据图示判断。每个标记角自己分类。)
- Q10. In each of the following diagrams, three lines intersect at a point. Find the value of $x$.
(a) Two angles $(5x-2)°$ and one of $84°$ with $(5x-2)°$ on top, $84°$ on st. line: → $x = $ 检查
(b) Three lines, angles labelled $x°$, $2x°$, $2x°$ on one side: → $x = $ 检查💡 思路
(a) 三角 (5x-2), 84, (5x-2) 在直线上 → $2(5x-2) + 84 = 180 \Rightarrow 10x = 100 \Rightarrow x = 10$.
(b) $x + 2x + 2x = 180 \Rightarrow 5x = 180 \Rightarrow x = 36$. - Q11. Lines $AD$, $BE$, $CF$ intersect at $G$. $\angle AGF = 38°$, $\angle EGD = 55°$. Find $\angle FGE = a°$, $\angle DGC = b°$, $\angle CGB = c°$.
$a = $ ; $b = $ ; $c = $ 检查💡 思路
$AD$ 是直线，3 个角加起来 = 180°：$38 + a + 55 = 180 \Rightarrow a = 87°$。
$b$ 与 $\angle AGF = 38°$ 对顶 →？等等，$b$ 是 $\angle DGC$。检查角对位：$BE$ 直线两端 $B$ 和 $E$；$CF$ 两端 $C, F$。
$\angle DGC$ 在 $b$ 的位置是 $\angle EGD$ 对顶 → $b = \angle EGD = 55°$.
$c = \angle CGB$ 与 $\angle AGF = 38°$ 对顶 → $c = 38°$. - Q12. The diagram shows a pair of tongs (two crossed bars). Angles at the pivot: $2x°$ (top), $7x°$ (left), $y°$ (right). Find $x$ and $y$.
$x = $ ; $y = $ 检查💡 思路
钳子是两条相交的直线，4 个角，$2x$ 和 $7x$ 相邻（在直线上）：$2x + 7x = 180 \Rightarrow x = 20$. $y$ 与 $7x$ 对顶 → $y = 7(20) = 140°$.
- Q13. The time shown on the clock is 4 o'clock. Find reflex $\angle AOB$ (where $A$ is at the hour hand, $B$ at the minute hand). → ° 检查
💡 思路
钟面平均分 12 区，每区 $360°/12 = 30°$。4 点钟时分针在 12，时针在 4，相隔 4 个区 → 顺时针小角度 = $4 \times 30° = 120°$。反射角 = $360° - 120° = 240°$.
- Q14. Two angles are vertically opposite angles. What is the size of each angle if the two angles are
(a) complementary angles (sum 90°)? → ° 检查
(b) supplementary angles (sum 180°)? → ° 检查💡 思路
对顶角相等，所以两个角相等。$x + x = 90 \Rightarrow x = 45°$. 或 $x + x = 180 \Rightarrow x = 90°$.
- Q15. $\angle x$ and $\angle y$ are the only two adjacent angles on a straight line. Find $\angle x$ if
(a) $\angle x = \angle y$. → ° 检查
(b) $\angle x = 3\angle y$. → ° 检查💡 思路
$x + y = 180$. (a) $2x = 180 \Rightarrow x = 90°$. (b) $3y + y = 180 \Rightarrow y = 45°, x = 135°$.
- Q16. In the diagram, $\angle PQT = 2\angle RQS$ and $\angle TQS = 5\angle RQS$. Write an equation in $x$ and solve it to find the size of $\angle TQS$. ($PQR$ is a straight line with $T$, $S$ as additional rays from $Q$.)
$\angle TQS = $ ° 检查💡 思路
Let $\angle RQS = r$. Then $\angle PQT = 2r$, $\angle TQS = 5r$. $PQR$ 直线 → $2r + 5r + r = 180 \Rightarrow 8r = 180 \Rightarrow r = 22.5°$. $\angle TQS = 5 \times 22.5 = 112.5°$.
- Q17. It is given that $OB \perp OD$ and $\angle AOB = \angle COD$. What is the relationship between $OA$ and $OC$?
Relationship: 检查💡 思路
$\angle BOD = 90°$. $\angle AOC = \angle AOB + \angle BOC$；$\angle BOC = \angle BOD - \angle COD = 90 - \angle COD$. 又 $\angle AOC = \angle AOB + (90 - \angle COD) = 90 + (\angle AOB - \angle COD) = 90 + 0 = 90°$. 所以 $OA \perp OC$.
- Q18. In the diagram, $AD$, $BE$, $CF$ are straight lines which intersect at $G$. Angles: $x°$ (between $FG$ and $AG$, top-left), $y°$ (between $BG$ and $AG$, bottom-left), $50°$ (between $EG$ and $DG$).
(a) Find $x° + y°$. → ° 检查
(b) Suggest two possible pairs of values of $x$ and $y$. 检查
(c) If $y = 3x$, find $x$ and $y$. → $x = $ ; $y = $ 检查💡 思路
(a) $AD$ 直线，3 角同侧加起来 180°：$x + 50 + y = 180 \Rightarrow x + y = 130°$.
(b) 任意满足和为 130 的对，比如 $(30, 100), (50, 80), (60, 70)$...
(c) $3x + x = 130 \Rightarrow 4x = 130 \Rightarrow x = 32.5°, y = 97.5°$. - Q19. (a) A circular pizza is divided into 10 equal pieces by 5 cuts through its centre. Find $\angle x$ (one piece). → ° 检查
(b) Another pizza is divided into equal pieces with each piece = $22.5°$. How many cuts are made?
Cuts = 检查💡 思路
(a) 360° / 10 = 36°.
(b) 360 / 22.5 = 16 块；每刀切出 2 块新区 → 16 / 2 = 8 刀。