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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
Year 8 · 8.15 Parallel Lines and Polygons

Solutions · Full Answer Key

Pack A answers · Pack B answers · Problem-solving worked solutions

Pack A — Answers

Bronze
1.47° (vertically opposite), 133°, 133°
2.75°
3.62°
4.x=70°x = 70°
5.720°
6.120°
7.60°
8.40°
9.130°
10.135°
Silver
11.58°
12.x=55x = 55
13.55° each
14.12 sides
15.10 sides
16.75°, 105°, 105° at upper; 75°, 105°, 105°, 75° at lower (mirrored)
17.x=135°x = 135°
18.140°
19.b=115°b = 115°; alternate = 115°
20.x=80°x = 80°
Gold
21.x=27x = 27
22.12 sides; sum 1800°
23.Obtuse co-int = 115°; acute on lower = 65°
24.x=95°x = 95°
25.b=110°,c=70°,d=110°b = 110°, c = 70°, d = 110°
26.15 sides; sum 2340°
27.72°,96°,120°,144°,168°72°, 96°, 120°, 144°, 168°
28.(a) 150° (b) 30° (c) 1800°
29.180° (forms a straight line: 60° triangle + 120° hexagon = 180°)
30.12 sides
Platinum
31.See working — a=ba = b via vertically opposite + corresponding
32.36°
33.x=131.25°x = 131.25°
34.x≈47.1°x \approx 47.1°
35.(a) Limit 180° (b) n≥12n \geq 12 (interior 150 for n=12n = 12, > 150 for n>12n > 12)
36.Sum of all four = 360°; two right angles → remaining two = 180° (supplementary)
37.8 sides
38.E.g. triangle (60°) + square (90°) + 12-gon (150°) — but 60+90+150 = 300, not 360. Try: hexagon (120°) × 3 → 360° ✓.
39.x=51.4°x = 51.4°; angles ≈ 51.4°, 102.9°, 41.4°, 164.3°
40.x=86.67°x = 86.67°

Pack B — Answers

Bronze
1.113° (vert opp), 67°, 67°
2.108°
3.144°
4.x=107°x = 107°
5.1080°
6.144°
7.36°
8.60°
9.105°
10.Same.
Silver
11.41°
12.x=80x = 80
13.65° each
14.8 sides
15.12 sides
16.Same with 60° / 120°.
17.Same.
18.≈ 147.3°
19.Same.
20.x=70°x = 70°
Gold
21.x=36x = 36
22.20 sides; sum 3240°
23.Obtuse 132°; acute 48°
24.x=117.5°x = 117.5°
25.Same.
26.20 sides; sum 3240°
27.54°,81°,108°,135°,162°54°, 81°, 108°, 135°, 162°
28.Same.
29.150° (90° + 60°)
30.Same.
Platinum
31.Same.
32.Same.
33.x=140°x = 140°
34.x=50°x = 50°
35.Same.
36.Same.
37.12 sides
38.Same.
39.Same.
40.x=78°x = 78°

Problem-solving — Worked Solutions

1Problem 1
Answer
(a) 5x+60=180,x=245x + 60 = 180, x = 24 (b) 92° and 88°; sum 180° ✓ (c) 92°
Full working
(a) (3x+20)+(2x+40)=180⇒5x=120⇒x=24(3x + 20) + (2x + 40) = 180 \Rightarrow 5x = 120 \Rightarrow x = 24.

(b) Angles: 3(24)+20=92°3(24) + 20 = 92°; 2(24)+40=88°2(24) + 40 = 88°. Sum 180° ✓.

(c) Corresponding = 92°.
2Problem 2
Answer
(a) 18° (b) 20 sides (c) 3240°
Full working
(a) Ext = 180−162=18°180 - 162 = 18°.

(b) Sides = 360/18=20360/18 = 20.

(c) Sum =(20−2)×180=3240°= (20 - 2) \times 180 = 3240°.
3Problem 3
Answer
(a) 108° (b) 36° (c) 5×36°=180°5 \times 36° = 180°
Full working
(a) Sum =540°= 540°. Each =108°= 108°.

(b) Each tip-triangle is isosceles with two base angles =180−108=72°= 180 - 108 = 72°. Tip =180−144=36°= 180 - 144 = 36°.

(c) 5×36°=180°5 \times 36° = 180°. (General result for any regular polygon star.)
4Problem 4
Answer
(a) x=30x = 30 (b) 70° each (c) 110°
Full working
(a) Vertically opposite angles are equal: 2x+10=3x−20⇒x=302x + 10 = 3x - 20 \Rightarrow x = 30.

(b) Both are 70°70°.

(c) Co-interior on ℓ2\ell_2: 180−70=110°180 - 70 = 110°.
5Problem 5
Answer
(a) Pentagon 108°, triangle 60° (b) 168° (c) No — 168° ≠ 360° per pair
Full working
(a) Pentagon 108°108°, triangle 60°60°.

(b) 108+60=168°108 + 60 = 168°.

(c) Tiling requires angle around a point = 360°. 108+60=168108 + 60 = 168, and other combinations don't reach 360 exactly without using more shapes. So a pure pentagon + triangle tiling doesn't work.
6Problem 6
Answer
Each interior + ext = 180°. Sum of interiors =(n−2)×180= (n-2) \times 180. Sum of all interior + ext = n×180n \times 180. So sum of exterior = n×180−(n−2)×180=2×180=360°n \times 180 - (n-2) \times 180 = 2 \times 180 = 360° ✓
Full working
Each vertex has one interior + one exterior = 180°. Sum over all nn vertices: n×180°n \times 180°. Subtract sum of interiors (n−2)×180(n-2) \times 180: exterior sum =n×180−(n−2)×180=2×180=360°= n \times 180 - (n-2) \times 180 = 2 \times 180 = 360°. ∎

This holds **for any convex polygon**, regular or not.
7Problem 7
Answer
(a) 120° (b) 332a2\tfrac{3\sqrt{3}}{2} a^2 (c) 6a6a
Full working
(a) Sum 720°720°; each 120°120°.

(b) Each equilateral triangle has area 34a2\tfrac{\sqrt{3}}{4} a^2. Six of them: 634a2=332a2\tfrac{6\sqrt{3}}{4} a^2 = \tfrac{3\sqrt{3}}{2} a^2.

(c) Perimeter 6a6a.
8Problem 8
Answer
(a) b=120°,c=60°,d=120°b = 120°, c = 60°, d = 120° (b) e=60°,f=120°,g=60°,h=120°e = 60°, f = 120°, g = 60°, h = 120° (c) See working
Full working
Upper: vertical pairs equal: a=c=60a = c = 60; b=d=120b = d = 120.

Lower: corresponding angles equal: e=a=60e = a = 60, etc. So e=60°,f=120°,g=60°,h=120°e = 60°, f = 120°, g = 60°, h = 120°.

Relationships: alternate aa and gg (both interior, on opposite sides); corresponding aa and ee; co-interior aa and ff; vertically opposite aa and cc, etc.
9Problem 9
Answer
(a) x=105°x = 105° (b) Convex (all < 180°) (c) Concave
Full working
(a) Sum = 540. x=540−100−110−95−130=105°x = 540 - 100 - 110 - 95 - 130 = 105°.

(b) All interior angles < 180° → convex.

(c) If any interior angle > 180° (reflex), the polygon is concave (has a "dent").
10Problem 10
Answer
(a) 60, 90, 120 all divide 360 (b) 108° does not (c) E.g. an irregular pentagon with specific angle measures
Full working
(a) Triangle 60° → 360/60=6360/60 = 6 triangles per vertex. Square 90° → 4. Hexagon 120° → 3.

(b) Pentagon 108°. 360/108=3.33360/108 = 3.33 — not an integer. So pure regular pentagon tiling impossible.

(c) Irregular pentagons with specific angle combinations can tile (e.g. the Cairo tiling using irregular convex pentagons).
11Problem 11
Answer
Use the fact that a+b+c+d=360°a + b + c + d = 360° at the intersection and use alternate / vertically opposite identities.
Full working
Let the angle at ℓ1\ell_1 on the "inside" of the parallel strip be α\alpha, and the co-interior angle at ℓ2\ell_2 be β\beta. The alternate angle to α\alpha at ℓ2\ell_2 (also between the parallels) is also α\alpha (alternate angles equal). And α\alpha and β\beta are now on a straight line at ℓ2\ell_2, so α+β=180°\alpha + \beta = 180°. ∎
12Problem 12
Answer
(a) 10 (b) 1440° (c) 36° (d) Interior increases with nn
Full working
(a) Ext = 36°. Sides = 360/36=10360/36 = 10.

(b) Sum =8×180=1440°= 8 \times 180 = 1440°.

(c) 36° each.

(d) Interior angle = 180−360/n180 - 360/n. As nn grows, 360/n360/n shrinks, so interior grows. For n+5>nn + 5 > n: interior at n+5n + 5 > interior at nn. ✓