Form 3 maths

Geometry and data basics

The junior geometry and statistics that later DSE questions rely on: 3-D figures, angles and parallel lines, polygons, congruent triangles, Pythagoras' theorem, and organising and presenting data. These give the reasons students write in geometry proofs and the chart-reading skills used in statistics questions.

Key terms

English	中文
vertically opposite angles	對頂角
corresponding / alternate / interior angles	同位角／內錯角／同旁內角
exterior angle of a triangle	三角形的外角
regular polygon	正多邊形
congruent triangles	全等三角形
isosceles triangle	等腰三角形
Pythagoras' theorem	畢氏定理
hypotenuse	斜邊
frequency distribution table	頻數分佈表
class interval, class mark	組區間，組中點
histogram	直方圖
cumulative frequency	累積頻數
stem-and-leaf diagram	幹葉圖

Standard steps

Find an angle and give the reason

Mark every angle you already know on the diagram.
Find one new angle at a time. After each one, write the reason in brackets.
Common reasons, in the short form exam markers accept:

Fact	English short form	中文
Angles on a straight line add to 180°	adj. ∠s on st. line	直線上的鄰角
Angles at a point add to 360°	∠s at a pt.	同頂角
Vertically opposite angles are equal	vert. opp. ∠s	對頂角
Corresponding angles, parallel lines	corr. ∠s, AB // CD	同位角，AB // CD
Alternate angles, parallel lines	alt. ∠s, AB // CD	內錯角，AB // CD
Interior angles add to 180°	int. ∠s, AB // CD	同旁內角，AB // CD
Angle sum of a triangle	∠ sum of △	三角形內角和
Exterior angle of a triangle	ext. ∠ of △	三角形外角
Base angles of an isosceles triangle	base ∠s, isos. △	等腰三角形底角

Check the short forms against the student's own textbook; publishers differ slightly.

Prove two triangles congruent

Name the triangles with matching vertices in the same order (△ABC ≅ △DEF).
List three pairs of equal parts, one per line, each with a reason (given, common side, vert. opp. ∠s, …).
State the condition: SSS, SAS, ASA, AAS or RHS.
Never use AAA or SSA. They do not prove congruence.

Polygon angles

Interior angle sum = (n − 2) × 180°.
Each exterior angle of a regular n-sided polygon = 360° ÷ n.
Interior + exterior = 180°.

Pythagoras

Find the hypotenuse (the side opposite the right angle).
Hypotenuse² = sum of the squares of the other two sides.
Converse: if c² = a² + b², the angle opposite c is 90°.

Read a cumulative frequency curve

The median is at half the total frequency.
Lower quartile at ¼, upper quartile at ¾ (used in F6).
Draw a horizontal line from the frequency axis to the curve, then down to read the value.

Worked examples

Example 1: Polygon angles: each interior angle of a regular polygon is 144°. How many sides does it have?

exterior angle = 180° − 144° = 36°
n = 360° ÷ 36° = 10

Example 2: Pythagoras' converse: a triangle has sides 5, 12 and 13. Is it right-angled?

5² + 12² = 25 + 144 = 169 = 13². Yes; by the converse of Pythagoras' theorem, the angle opposite the side 13 is 90°.

Example 3: Pythagoras: a 5 m ladder leans against a wall with its foot 1.4 m from the wall. How high up the wall does it reach?

the ladder is the hypotenuse (opposite the right angle at the wall's foot)
h² + 1.4² = 5²
h²        = 25 − 1.96 = 23.04
h         = 4.8 m

Example 4: Find an angle and give the reason: AB // CD, and a line meets AB at P and CD at Q, with A and D on opposite sides of PQ. ∠APQ = 65°. Find ∠PQD and ∠PQC.

∠PQD = ∠APQ = 65°           (alt. ∠s, AB // CD)
∠PQC = 180° − 65° = 115°    (adj. ∠s on st. line)

Example 5: Prove two triangles congruent: AB and CD cross at M, with AM = BM and CM = DM. Prove △AMC ≅ △BMD.

AM = BM            (given)
∠AMC = ∠BMD        (vert. opp. ∠s)
CM = DM            (given)
∴ △AMC ≅ △BMD      (SAS)

The angle is between the two sides in both triangles, so this is SAS, not SSA.

Example 6: Read a cumulative frequency curve: the curve for 80 students' marks passes through (52, 20), (61, 40) and (70, 60). Find the median and the interquartile range.

median: ½ × 80 = 40th student  → 61 marks
Q1:     ¼ × 80 = 20th student  → 52 marks
Q3:     ¾ × 80 = 60th student  → 70 marks
interquartile range = 70 − 52 = 18 marks

Common mistakes

Writing "corr. ∠s" without saying which lines are parallel.
Using "alternate angles" when the lines are not parallel.
Naming congruent triangles in the wrong vertex order, so later equal sides don't match.
Using SSA ("two sides and a non-included angle").
Treating a histogram's bar heights as frequencies when class widths differ (the DSE usually uses equal widths).

How the DSE tests it

Geometry proofs in A(1) often start with congruent triangles.
Paper 2 has angle questions that combine parallel lines, isosceles triangles and polygons.
Statistics questions start from a stem-and-leaf diagram, frequency table or cumulative frequency curve.

Questions you might not ask in class

Why do I have to write the reason if the answer is right?

In a proof, the reasons are the answer. Markers give marks for each correct reason.

Why doesn't AAA prove congruence?

Two triangles can have the same angles but different sizes, like a photo and its enlargement. AAA proves similar, not congruent.

How do I know which angle facts to use?

Look for parallel lines first, then isosceles triangles, then the triangle angle sum. Try one; if it gives a new angle, keep going.