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Chapter 7 · Review & Problem-Solving

📚 LET'S SUM UP!

Polygons

A polygon is a closed plane figure formed by three or more line segments.
A polygon is classified according to its number of sides (triangle 3, quad 4, pentagon 5, hexagon 6, ...).

Triangles (3-sided polygons)

In any $\triangle ABC$ with sides $a$ (opp. $A$), $b$, $c$ and angles $\angle A$, $\angle B$, $\angle C$:

Triangle inequality: $AB + AC > BC$ (and the other two combinations).
If $\angle BAC$ is the largest angle, then $BC$ is the longest side.
Sum of interior angles: $\angle a + \angle b + \angle c = 180°$. ∠ sum of △
Exterior angle = sum of two opposite interior angles: $\angle a + \angle b = $ ext. ∠ at $C$. ext. ∠ of △

Quadrilaterals (4-sided polygons) & Other Polygons

Six special quadrilaterals: parallelogram, rectangle, rhombus, square, trapezium, kite. Hierarchy:

At least 1 pair of parallel sides → Trapezium
2 pairs of parallel sides → Parallelogram → (with 4 right ∠s) Rectangle → (with 4 equal sides) Square
2 pairs of parallel sides + 4 equal sides → Rhombus → (with 4 right ∠s) Square
2 pairs of equal adjacent sides + diagonals perpendicular → Kite

Polygons

An $n$-sided polygon is also called an $n$-gon.
Sum of interior angles of an $n$-gon = $(n-2) \times 180°$. ∠ sum of polygon
Sum of exterior angles of any convex polygon = $360°$. ext. ∠ sum of polygon
A regular $n$-gon has each interior $= \dfrac{(n-2) \cdot 180°}{n}$ and each exterior $= \dfrac{360°}{n}$.

Symmetry Properties of Polygons

A line of symmetry divides an object into two identical halves. An object may have more than one line of symmetry.
The order of rotational symmetry is the number of times an object fits onto itself in one rotation of $360°$.
A regular $n$-gon has exactly $n$ lines of symmetry and rotational symmetry of order $n$.

RReview Exercise 7

Q1. Find $x$.
(a) $\triangle PRQ$: $\angle R = 1.5x°$, $\angle Q = 2x°$, $\angle P = x°$. $x = $ $°$ 检查
(b) 图中 $5x°$、$3x°$、$4x°$ 是同一个三角形的三个外角（在每条边延长形成的外角处分别标注）。Find $x$. $x = $ $°$ 检查
💡 思路
(a) $\triangle PRQ$ 内角和 $= 180°$：$x + 1.5x + 2x = 4.5x = 180° \Rightarrow x = 40°$。
(b) 三角形的三个外角之和恒为 $360°$（Practice 7.1 Q12 已证）。所以 $5x + 3x + 4x = 12x = 360° \Rightarrow x = 30°$。
Q2. Find $x$ and $y$ in each diagram.
(a) $x = $ $°$, $y = $ $°$ 检查 x 检查 y
(b) 5-point star: $x = $ $°$, $y = $ $°$ 检查 x 检查 y
Q3. Rhombus $ABCD$ with $AD$ produced to $E$, $\triangle CDE$ equilateral. Find $\angle BEC$.
$\angle BEC = $ $°$ 检查
💡 思路
$\triangle CDE$ equilateral → $DE = DC$, $\angle CDE = 60°$. On line $A$-$D$-$E$: $\angle ADC + 60° = 180° \Rightarrow \angle ADC = 120°$. So rhombus has $\angle ADC = 120°$, $\angle BCD = 60°$. $\triangle BCE$: $BC = CD = CE$ (rhombus side = equilateral side), $\angle BCE = \angle BCD + \angle DCE = 60+60 = 120°$. Isos with apex $120°$ → base angles $= 30°$. So $\angle BEC = 30°$.
Q4. $AOD$, $BOE$, $COF$ are straight lines through $O$. Find $a + b + c + d + e + f$.
$a + b + c + d + e + f = $ $°$ 检查
💡 思路
Three lines through $O$ form 6 angles around $O$, and the sum of angles at a point is $360°$.
Q5. Parallelogram $PQRS$ with $QR$ produced to $V$ and $SR$ produced to $T$. $\angle VRU = 26°$ ($U$ on extension), $\angle SPQ = 73°$. Find $x$ and $y$.
$x = $ $°$, $y = $ $°$ 检查 x 检查 y
Q6. Rectangle $ABCD$ with diagonals intersecting at $E$. $\triangle EDF$ equilateral (where $F$ is a point on a diagonal-related construction), $\angle ABD = 60°$.
(a) $x = $ $°$, $y = $ $°$, $z = $ $°$ 检查 x 检查 y 检查 z
(b) If $CE = 3$ cm, $BD = $ cm 检查
(c) Is $AEDF$ a rhombus? 检查
💡 思路
Diagonals of rectangle bisect & equal → $AE = BE = CE = DE$. Tri $EDF$ equilateral → $EF = ED = FD$. With $\angle ABD = 60°$, deduce other angles. $BD = 2 \cdot CE = 6$. $AEDF$ has $AF = FD = DE = EA$ (4 equal sides from equilateral structure) → rhombus.
Q7. Find $x$ in each diagram.
(a) Pentagon with various angles → $x = $ $°$ 检查
(b) Hexagon with various angles → $x = $ $°$ 检查
Q8. Construct quadrilateral $ABCD$ with $AB = 3$ cm, $BC = 3.5$ cm, $CD = 2.5$ cm, $DA = 3$ cm, $BD = 4$ cm. (No numeric answer.)
Q9. Regular pentagon $ABCDE$ and equilateral $\triangle FCD$ (with $F$ outside, attached to side $CD$). Find $x$ and $y$.
$x = $ $°$, $y = $ $°$ 检查 x 检查 y
Q10. Regular hexagon $ABCDEF$ and regular octagon $ABPQRSTU$ sharing side $AB$.
(a) $\angle CBP = $ $°$ 检查
(b) $\angle CAP = $ $°$ 检查
💡 思路
(a) At $B$: $\angle ABC + \angle ABP + \angle CBP = 360°$. Hex interior $= 120°$, oct interior $= 135°$. $\angle CBP = 360 - 120 - 135 = 105°$.
Q11. In an $n$-gon, ratio sum-int : sum-ext $= 10 : 1$. Find $n$.
$n = $ 检查
💡 思路
$(n-2) \cdot 180 / 360 = 10 \Rightarrow (n-2) = 20 \Rightarrow n = 22$.
Q12. Each interior angle of a regular polygon is $157.5°$. Find number of sides.
$n = $ 检查
💡 思路
Each ext $= 180-157.5 = 22.5°$. $n = 360/22.5 = 16$.
Q13. $ABCD$ is part of a regular polygon with $\angle ABC = 144°$. $AB$ and $DC$ are produced to meet at $E$.
(a) Number of sides: 检查
(b) $\angle BEC = $ $°$, $\angle CDB = $ $°$ 检查 BEC 检查 CDB
💡 思路
(a) Each ext $= 180-144 = 36°$; $n = 360/36 = 10$.
(b) Tri $BEC$: $\angle EBC = \angle ECB = 36°$ (ext angles at $B$ and $C$); $\angle BEC = 180 - 36 - 36 = 108°$.
Q14. $AB$, $BC$, $CD$ three consecutive sides of regular 15-gon. Lines $AC$, $DB$ intersect at $P$. Find $x = \angle APD$.
$x = $ $°$ 检查
💡 思路
正 15 边形每个内角 $= (15-2) \cdot 180°/15 = 156°$。
$\triangle ABC$ 等腰（$AB = BC$）→ 底角 $\angle BAC = \angle BCA = (180° - 156°)/2 = 12°$。
同理 $\triangle BCD$（$BC = CD$）→ $\angle CBD = \angle BDC = 12°$。
$P$ 是 $AC$ 与 $BD$ 的交点 → 在 $\triangle BPC$ 内：$\angle BPC = 180° - \angle PBC - \angle PCB = 180° - 12° - 12° = 156°$。
最后 $\angle APD = \angle BPC = 156°$（vert. opp. ∠s）。即 $\boxed{x = 156°}$。
Q15. What is the maximum number of interior acute angles a polygon can have?
Max acute angles = 检查
💡 思路
Sum of exterior angles $= 360°$. Each acute interior corresponds to an exterior $> 90°$. If $k$ vertices are acute interiors, sum of those $k$ exteriors $> 90k$. We need $90k < 360$, so $k < 4 \Rightarrow k \le 3$. Three is achievable (e.g. a triangle with all acute angles, though the question asks across all polygon types — the limit holds).

🧠 Problem-Solving Task · 蜜蜂铺砖 (Tessellation)

Bees build honeycomb walls of identical regular hexagons that fit together with no gaps or overlaps — a tessellation. Which regular polygons (apart from hexagons) can form a tessellation?

Use Pólya's 4-step problem-solving process:

STEP 1. Understanding the Problem
• Why is it possible for regular hexagons to form a tessellation?
• Can all regular polygons form a tessellation?
• What is the goal of this problem?

STEP 2. Devising a Plan
• What is the interior angle of an $n$-sided regular polygon in terms of $n$?
→ $(n-2) \cdot 180° / n$
• For a tessellation, what condition must the interior angle satisfy?
→ Some integer multiple of it must equal $360°$.

STEP 3. Carrying Out the Plan

$n$ (regular)	Interior $= (n-2) \cdot 180/n$	$360 / $ int	Integer? Tiles?
3 (tri)	$60°$	$6$	YES ✓
4 (square)	$90°$	$4$	YES ✓
5 (pent)	$108°$	$3.33...$	NO ✗
6 (hex)	$120°$	$3$	YES ✓
7 (hept)	$128.57°$	$2.8$	NO ✗
$n \ge 7$	int $> 120°$	between 2 and 3	never integer

$\Rightarrow$ Only 3 regular polygons tessellate alone: equilateral triangle, square, regular hexagon.

STEP 4. Looking Back & Extension
• Is the answer reasonable? Yes — bees pick the hexagon because it gives the most floor area per unit of wall (proven separately).
• Extension: given any two regular polygons of $x$ and $y$ sides, can they tessellate together? (E.g. octagons + squares tessellate; triangles + hexagons tessellate.)

📔 Maths Journal

(1) Write some notes to classify quadrilaterals.

(2) What are the invariant properties you have learnt in this chapter?