Answer Key

8.14 Triangles

Pack A — Answers

#	Question	Answer
1	A triangle has angles 65° and 80°. Find the third angle.	35°
2	An isosceles triangle has apex angle 40°. Find each base angle.	70°
3	In triangle ABC, $\widehat{ABC} = 50°$ and $\widehat{BCA} = 70°$. Find $\widehat{BAC}$.	60°
4	Classify the triangle with sides 3, 4, 5 cm.	Scalene right-angled triangle
5	Can a triangle have sides 4, 5, 8 cm? Justify using the triangle inequality.	Yes — 4+5 = 9 > 8 ✓
6	State the uniqueness conditions for triangles (SSS, SAS, ASA, RHS).	3 sides; 2 sides + included angle; 2 angles + 1 side; right angle + hypotenuse + 1 side
7	In an equilateral triangle, what is each interior angle?	60°
8	A triangle has angles 90°, 35°, $x$. Find $x$.	55°
9	Classify by sides: a triangle has two sides of equal length. State the name.	Isosceles
10	Classify by angles: a triangle has one angle of 120°. State the type.	Obtuse
11	A triangle has angles $(2x + 10)°$, $(3x - 5)°$ and $90°$. Find $x$.	$x = 17$
12	In triangle ABC, sides $a, b, c$ correspond to vertices $A, B, C$. If $a = 6$, $b = 8$, $c = 10$, identify which is the longest, the smallest, and find the right angle (if any).	Longest c (10); smallest a (6). Right-angle at $C$ ($6^2 + 8^2 = 10^2$).
13	A triangle has sides 7, 9, $x$ cm. The triangle inequality requires what range for $x$?	$2 < x < 16$
14	A triangle is isosceles with base 8 cm and equal sides 6 cm each. Calculate (a) the perimeter, (b) the height from the apex to the base, (c) the area.	(a) 20 cm (b) $\sqrt{36 - 16} = \sqrt{20} \approx 4.47$ cm (c) ≈ 17.9 cm²
15	Two triangles have sides 3, 4, 5 and 5, 12, 13. Are they congruent? Similar?	Neither — sides not proportional
16	Two triangles have sides 6, 8, 10 and 12, 16, 20. Are they similar?	Yes — all sides scaled by 2
17	The exterior angle of a triangle equals the sum of the two non-adjacent interior angles. If interior angles are 40° and 60°, find the exterior angle at the third vertex.	100°
18	A right-angled triangle has legs 5 and 12. Find the hypotenuse.	13
19	A right-angled triangle has hypotenuse 25 and one leg 7. Find the other leg.	24
20	A triangle has sides labelled $a$, $b$, $c$. State which side is opposite vertex $A$ and explain the naming convention.	Side $a$ is opposite vertex $A$. Lowercase letter matches uppercase vertex.
21	In a figure with two parallel lines and a transversal, one angle is $(3x - 10)°$ and the corresponding angle on the other line is $(2x + 20)°$. Find $x$.	$x = 30$
22	Triangle ABC has $A = (0, 0)$, $B = (4, 0)$, $C = (4, 3)$. (a) Find the perimeter. (b) Classify by sides and angles.	(a) 12 (b) Right-angled scalene
23	A triangle has angles $(2x)°, (3x + 10)°$ and $(x - 10)°$. Find $x$ and the three angles.	$x = 30$; angles 60°, 100°, 20°
24	An isosceles triangle has perimeter 30 cm and base 8 cm. Find the equal sides.	11 cm each
25	A triangle has angles $(x + 20)°$, $(2x - 10)°$, $(3x)°$. (a) Find $x$. (b) Classify the triangle.	(a) $x = 28.3$ (b) Scalene
26	A 3D Pythagoras-flavoured problem: a rectangular room is 4 m × 3 m × 2 m. Find the longest diagonal.	$\sqrt{29} \approx 5.39$ m
27	Two triangles with the same sides (SSS) are always congruent. True or false? Justify.	True — three sides uniquely determine a triangle's shape and size
28	A 2D figure on a coordinate plane has vertices $A(0,0), B(6,0), C(3,4)$. Show that this is an isosceles triangle and find the height from $C$ to $AB$.	AC = BC = 5; isosceles. Height = 4.
29	A right-angled triangle has legs $a$ and $b$ and hypotenuse $c$, with $a:b = 3:4$. If the perimeter is 60 cm, find each side.	15, 20, 25 cm
30	Triangle ABC has vertices at $A(1,2), B(5,2), C(3,6)$. Show that ABC is isosceles and find its area.	AC = BC; area = 8 sq units
31	A triangle has sides 7, 24, 25. (a) Verify it is right-angled. (b) Find the area. (c) Find the height from the right angle to the hypotenuse.	(a) $7^2 + 24^2 = 49 + 576 = 625 = 25^2$ ✓ (b) 84 (c) 6.72
32	Two similar triangles have sides in ratio 3:4. If the smaller has perimeter 21 cm, find the larger's perimeter. If the smaller has area 9 cm², find the larger's area.	Perimeter 28 cm; area 16 cm²
33	A triangle ABC has $A(0,0), B(8,0), C(4,6)$. (a) Find each side. (b) Find each angle.	(a) AB = 8, BC = AC = $\sqrt{52}$ (b) angles at A = B ≈ 56.3°; at C ≈ 67.4°
34	A triangle has angles $A$, $B$, $C$ where $A:B:C = 2:3:5$. Find the angles.	36°, 54°, 90°
35	Two triangles have angles (40°, 60°, 80°) and (40°, 60°, 80°). Sides of the first are 5, 6, 8. Sides of the second are 10, 12, 16. (a) Show they are similar. (b) Find the ratio of areas.	Similar (AAA + sides scale by 2); area ratio 1 : 4
36	A triangle has area 30 cm² and base 10 cm. Find the height. Then find the angles if the triangle is isosceles with the height bisecting the base.	Height 6 cm. Base angles $\tan^{-1}(6/5) \approx 50.2°$; apex ≈ 79.6°
37	A right-angled triangle has hypotenuse 10 cm and legs $a$ and $b$, where $a + b = 14$. Find $a$ and $b$.	$a = 6, b = 8$ (or vice versa)
38	Verify the triangle inequality holds: a triangle has sides $a, b, c$ with $a + b > c$, $a + c > b$, $b + c > a$. If all three are equal, can the triangle still be valid?	Yes — equilateral triangle. The strict inequality holds for any positive equal sides.
39	A triangle has the angles in the ratio 1:1:2. (a) Find the angles. (b) Show that this is a right-angled isosceles triangle.	45°, 45°, 90°. Right-angled isosceles ✓
40	Two triangles have sides 4, 5, 6 and 6, 7.5, 9. (a) Are they similar? (b) If yes, find the linear and area scale factors.	Yes; linear sf 1.5; area sf 2.25

Pack B — Answers

#	Question	Answer
1	A triangle has angles 42° and 73°. Find the third angle.	65°
2	An isosceles triangle has apex angle 100°. Find each base angle.	40°
3	In triangle ABC, $\widehat{ABC} = 50°$ and $\widehat{BCA} = 70°$. Find $\widehat{BAC}$.	40°
4	Classify the triangle with sides 3, 4, 5 cm.	Scalene right-angled triangle
5	Can a triangle have sides 3, 4, 8 cm? Justify using the triangle inequality.	No — 3+4 = 7 < 8
6	State the uniqueness conditions for triangles (SSS, SAS, ASA, RHS).	Same.
7	In an equilateral triangle, what is each interior angle?	60°
8	A triangle has angles 90°, 35°, $x$. Find $x$.	40°
9	Classify by sides: a triangle has two sides of equal length. State the name.	Scalene
10	Classify by angles: a triangle has one angle of 120°. State the type.	Acute
11	A triangle has angles $(2x + 10)°$, $(3x - 5)°$ and $90°$. Find $x$.	$x = 30$
12	In triangle ABC, sides $a, b, c$ correspond to vertices $A, B, C$. If $a = 6$, $b = 8$, $c = 10$, identify which is the longest, the smallest, and find the right angle (if any).	Longest c (13); right angle at $C$.
13	A triangle has sides 7, 9, $x$ cm. The triangle inequality requires what range for $x$?	$6 < x < 16$
14	A triangle is isosceles with base 8 cm and equal sides 6 cm each. Calculate (a) the perimeter, (b) the height from the apex to the base, (c) the area.	(a) 32 cm (b) 8 cm (c) 48 cm²
15	Two triangles have sides 3, 4, 5 and 5, 12, 13. Are they congruent? Similar?	Same.
16	Two triangles have sides 6, 8, 10 and 12, 16, 20. Are they similar?	Yes — scale factor 3
17	The exterior angle of a triangle equals the sum of the two non-adjacent interior angles. If interior angles are 40° and 60°, find the exterior angle at the third vertex.	120°
18	A right-angled triangle has legs 5 and 12. Find the hypotenuse.	17
19	A right-angled triangle has hypotenuse 25 and one leg 7. Find the other leg.	15
20	A triangle has sides labelled $a$, $b$, $c$. State which side is opposite vertex $A$ and explain the naming convention.	Same.
21	In a figure with two parallel lines and a transversal, one angle is $(3x - 10)°$ and the corresponding angle on the other line is $(2x + 35)°$. Find $x$.	$x = 45$
22	Triangle ABC has $A = (0, 0)$, $B = (4, 0)$, $C = (4, 3)$. (a) Find the perimeter. (b) Classify by sides and angles.	(a) 24 (b) Right-angled scalene
23	A triangle has angles $(2x)°, (3x + 10)°$ and $(x - 10)°$. Find $x$ and the three angles.	Same.
24	An isosceles triangle has perimeter 30 cm and base 8 cm. Find the equal sides.	14 cm each
25	A triangle has angles $(x + 20)°$, $(2x - 10)°$, $(3x)°$. (a) Find $x$. (b) Classify the triangle.	Same.
26	A 3D Pythagoras-flavoured problem: a rectangular room is 4 m × 3 m × 2 m. Find the longest diagonal.	$\sqrt{50} \approx 7.07$ m
27	Two triangles with the same sides (SSS) are always congruent. True or false? Justify.	Same.
28	A 2D figure on a coordinate plane has vertices $A(0,0), B(6,0), C(3,4)$. Show that this is an isosceles triangle and find the height from $C$ to $AB$.	AC = BC = $\sqrt{52}$; isosceles; height 6.
29	A right-angled triangle has legs $a$ and $b$ and hypotenuse $c$, with $a:b = 3:4$. If the perimeter is 60 cm, find each side.	15, 36, 39 cm
30	Triangle ABC has vertices at $A(1,2), B(5,2), C(3,6)$. Show that ABC is isosceles and find its area.	AC = BC = $\sqrt{34}$; area 15
31	A triangle has sides 7, 24, 25. (a) Verify it is right-angled. (b) Find the area. (c) Find the height from the right angle to the hypotenuse.	(a) $64 + 225 = 289 = 17^2$ ✓ (b) 60 (c) ≈ 7.06
32	Two similar triangles have sides in ratio 3:4. If the smaller has perimeter 21 cm, find the larger's perimeter. If the smaller has area 9 cm², find the larger's area.	Same.
33	A triangle ABC has $A(0,0), B(8,0), C(4,6)$. (a) Find each side. (b) Find each angle.	Same.
34	A triangle has angles $A$, $B$, $C$ where $A:B:C = 2:3:5$. Find the angles.	30°, 60°, 90°
35	Two triangles have angles (40°, 60°, 80°) and (40°, 60°, 80°). Sides of the first are 5, 6, 8. Sides of the second are 10, 12, 16. (a) Show they are similar. (b) Find the ratio of areas.	Same.
36	A triangle has area 30 cm² and base 10 cm. Find the height. Then find the angles if the triangle is isosceles with the height bisecting the base.	Same.
37	A right-angled triangle has hypotenuse 10 cm and legs $a$ and $b$, where $a + b = 14$. Find $a$ and $b$.	$a = 5, b = 12$
38	Verify the triangle inequality holds: a triangle has sides $a, b, c$ with $a + b > c$, $a + c > b$, $b + c > a$. If all three are equal, can the triangle still be valid?	Same.
39	A triangle has the angles in the ratio 1:1:2. (a) Find the angles. (b) Show that this is a right-angled isosceles triangle.	Same.
40	Two triangles have sides 4, 5, 6 and 6, 7.5, 9. (a) Are they similar? (b) If yes, find the linear and area scale factors.	Same.

Problems — Worked Solutions

**Triangle inequality - feasibility.** A student wants to construct a triangle with whole-number side lengths and perimeter 12 cm. (a) Use the triangle inequality to find all valid sets of three whole-number side lengths. (b) Which is the only equilateral solution? (c) Which sets give isosceles triangles? (d) Which set is the special 3-4-5 right-angle triple?

Answer

(a) {2, 5, 5}, {3, 4, 5}, {4, 4, 4} (b) {4, 4, 4} (c) {2, 5, 5}, {4, 4, 4} (d) {3, 4, 5}

Triangle inequality: any two sides > third side. Lists $a \leq b \leq c$: If $c \geq 6$: $a + b \leq 6$ violates inequality. So $c \leq 5$. $c = 5$: $a + b = 7$, with $a \leq b \leq 5$. Options: $(2,5), (3,4)$. $c = 4$: $a + b = 8$, $a \leq b \leq 4$. Only $(4,4)$. Valid sets: {2, 5, 5}, {3, 4, 5}, {4, 4, 4}.

**Constructing an isosceles triangle.** Construct an isosceles triangle with base 6 cm and equal sides 5 cm. (a) Describe the construction with compasses and straight-edge. (b) Calculate the triangle's height using Pythagoras. (c) Find its area.

Answer

(a) See working (b) Height = 4 cm (c) 12 cm²

(a) Draw AB = 6 cm. Open compasses to 5 cm. From A draw arc above AB. From B draw arc that intersects the first. Call intersection C. Join AC and BC. (b) Drop perpendicular from C to AB midpoint M. AM = 3, AC = 5. By Pythagoras: $CM = \sqrt{25 - 9} = 4$. (c) Area = $\tfrac{1}{2}(6)(4) = 12$ cm².

**Algebra in a triangle.** A triangle has angles $(2x)°$, $(3x + 10)°$, $(x - 10)°$. (a) Set up and solve. (b) Find the angles. (c) Classify the triangle by angles.

Answer

(a) $6x = 180, x = 30$ (b) 60°, 100°, 20° (c) Obtuse scalene

(a) Sum 180: $2x + 3x + 10 + x - 10 = 6x = 180 \Rightarrow x = 30$. (b) $60°, 100°, 20°$. (c) One angle > 90° → obtuse. All different → scalene.

**Pythagorean triples.** A right-angled triangle has legs $a$ and $b$, hypotenuse $c$. (a) Verify that $(3, 4, 5)$, $(5, 12, 13)$, $(8, 15, 17)$ are Pythagorean triples. (b) Generate a new triple by multiplying $(3, 4, 5)$ by 4. (c) Show that $(7, 24, 25)$ is a triple.

Answer

(a) All verified (b) (12, 16, 20) (c) $7^2 + 24^2 = 49 + 576 = 625 = 25^2$ ✓

(a) $9 + 16 = 25 = 5^2$ ✓. $25 + 144 = 169 = 13^2$ ✓. $64 + 225 = 289 = 17^2$ ✓. (b) $(12, 16, 20)$: $144 + 256 = 400 = 20^2$ ✓. (c) $49 + 576 = 625 = 25^2$ ✓. So $(7, 24, 25)$ is also a triple.

**Distance and triangle area.** A triangle has vertices $A(0, 0)$, $B(8, 0)$, $C(4, 6)$. (a) Find each side. (b) Find the perimeter. (c) Find the area. (d) Classify the triangle.

Answer

(a) AB = 8, BC = $\sqrt{52}$, AC = $\sqrt{52}$ (b) ≈ 22.4 (c) 24 (d) Isosceles

(a) AB = 8. AC = $\sqrt{16 + 36} = \sqrt{52} \approx 7.21$. BC = $\sqrt{16 + 36} = \sqrt{52}$. (b) $8 + 2\sqrt{52} \approx 22.4$. (c) Base 8, height 6: $A = \frac{1}{2}(8)(6) = 24$. (d) Two sides equal → isosceles. (Not right-angled since $52 + 52 = 104 \neq 64$.)

**Building uniqueness.** Two triangles are constructed with the data: A: sides 6, 8, 10. B: sides 6, 8 with included angle 90°. C: sides 6, 8 with included angle 45°. (a) Are A and B congruent? (b) Are A and C congruent? (c) State which uniqueness rules apply.

Answer

(a) Yes — by SAS and SSS (since C is the right angle) (b) No — different shape (c) SSS, SAS

Triangle A: 6, 8, 10 → right angle at the joint between legs 6 and 8 (Pythagoras: $36 + 64 = 100$). Triangle B: 6, 8 with included angle 90° → third side = $\sqrt{100} = 10$. Same as A. Triangle C: 6, 8 with included angle 45° → third side $\neq 10$. Different shape. (a) Yes — congruent (same SSS data). (b) No — different included angle gives different triangle. (c) Rules: **SSS** (A from sides 6, 8, 10), **SAS** (B from 6, 8, 90°). Both produce the same triangle.

**Triangle classification.** Classify each triangle by sides AND angles: (a) Sides 5, 5, 5 (b) Sides 3, 4, 5 (c) Sides 6, 8, 11 (d) Sides 7, 7, 10

Answer

(a) Equilateral, acute (b) Scalene, right (c) Scalene, obtuse (d) Isosceles, acute

(a) Equilateral (all sides equal); all 60° angles → acute. (b) Scalene; right ($3^2 + 4^2 = 5^2$). (c) Scalene; check $36 + 64 = 100 < 121 = 11^2$ → obtuse. (d) Isosceles (two equal sides); check $49 + 49 = 98 > 100 = 10^2$ → acute.

**Find the missing side.** A right-angled triangle has hypotenuse 17 and one leg 8. Find the other leg. Use the result to find a value of $x$ in the triangle with sides $x$, $x + 1$, $x + 17$ — assume this is right-angled.

Answer

Other leg = 15. For the parametric triangle: see working.

Pythagoras: $8^2 + b^2 = 17^2 \Rightarrow b^2 = 289 - 64 = 225 \Rightarrow b = 15$. Bigger: try $x = 8, x + 17 = 25, x + 1 = 9$ — check $64 + 81 = 145 \neq 625$. So $x = 8$ doesn't work for this parametric. For the second triangle: $x^2 + (x+1)^2 = (x+17)^2$. Expand: $2x^2 + 2x + 1 = x^2 + 34x + 289$. So $x^2 - 32x - 288 = 0$. Discriminant $= 1024 + 1152 = 2176$. $\sqrt{2176} \approx 46.6$. $x = (32 + 46.6)/2 \approx 39.3$. Not a clean integer — adjust parametric expectations.

**Angles in an isosceles triangle.** An isosceles triangle has equal sides of length 13 cm and base 10 cm. (a) Find the perpendicular height from apex to base. (b) Find the area. (c) Find the two base angles (using inverse tan). (d) Find the apex angle.

Answer

(a) 12 cm (b) 60 cm² (c) ≈ 67.4° (d) ≈ 45.2°

(a) Drop perpendicular from apex to midpoint of base. Right triangle: half-base = 5, hyp = 13. Height = $\sqrt{169 - 25} = 12$. (b) Area = $\tfrac{1}{2}(10)(12) = 60$ cm². (c) Each base angle: $\tan^{-1}(12/5) = \tan^{-1}(2.4) \approx 67.4°$. (d) Apex $= 180 - 2 \times 67.4 = 45.2°$.

**Outdoors application.** A 6 m vertical pole stands on horizontal ground. A wire is anchored from the top of the pole to a point on the ground 8 m from the base of the pole. (a) Sketch the triangle formed. (b) Find the length of the wire. (c) Find the angle the wire makes with the ground.

Answer

(a) Right triangle (b) 10 m (c) ≈ 36.9°

(a) Right-angled triangle with vertical leg 6, horizontal 8, hypotenuse (wire) unknown. (b) Wire = $\sqrt{36 + 64} = 10$ m. (c) Angle = $\tan^{-1}(6/8) = \tan^{-1}(0.75) \approx 36.9°$.

**Triangle similarity.** Two triangles ABC and DEF are similar with linear scale factor $k = 1.5$. (a) The smaller has sides 6, 8, 10. Find the sides of the larger. (b) Find the ratio of perimeters. (c) Find the ratio of areas.

Answer

(a) 9, 12, 15 (b) 1 : 1.5 (c) 1 : 2.25 (or 4 : 9)

(a) Multiply by 1.5: 9, 12, 15. (b) Perimeter ratio = linear sf = 1 : 1.5. (c) Area ratio = (linear sf)² = $1 : 2.25 = 4 : 9$.

**Investigating triangle types.** For each set of angles, decide if the triangle is acute / right / obtuse, and explain. (a) 30°, 60°, 90° (b) 50°, 60°, 70° (c) 100°, 40°, 40° (d) 89°, 89°, 2°

Answer

(a) Right (b) Acute (c) Obtuse (d) Acute (all < 90°)

(a) Has a 90° → right. (b) All < 90° → acute. (c) Has 100° → obtuse. (d) All < 90° (89, 89, 2) → acute, even though some are very close to 90°.