Triangle Inequality Theorem Worksheets - Math Monks - Free Printable

Let’s work through each part of the worksheet step by step.

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Problem 1: Decide whether each set of numbers can form a triangle

We use the Triangle Inequality Theorem:
> The sum of any two sides must be greater than the third side.
So for sides a, b, c:
- a + b > c
- a + c > b
- b + c > a

If all three are true → it’s a triangle.
If even one fails → not a triangle.

Let’s check each:

(a) 5, 10, 15
5 + 10 = 15 → NOT greater than 15 → ✘ Not a triangle

(b) 6, 9, 16
6 + 9 = 15 < 16 → ✘ Not a triangle

(c) 9, 40, 41
9 + 40 = 49 > 41 ✔
9 + 41 = 50 > 40 ✔
40 + 41 = 81 > 9 ✔ → ✔️ Triangle

(d) 7.5, 8.5, 14.5
7.5 + 8.5 = 16 > 14.5 ✔
7.5 + 14.5 = 22 > 8.5 ✔
8.5 + 14.5 = 23 > 7.5 ✔ → ✔️ Triangle

(e) 11, 12, 9
11 + 12 = 23 > 9 ✔
11 + 9 = 20 > 12 ✔
12 + 9 = 21 > 11 ✔ → ✔️ Triangle

(f) 1, 16, 16
1 + 16 = 17 > 16 ✔
1 + 16 = 17 > 16 ✔
16 + 16 = 32 > 1 ✔ → ✔️ Triangle

(g) 9, 5, 7
9 + 5 = 14 > 7 ✔
9 + 7 = 16 > 5 ✔
5 + 7 = 12 > 9 ✔ → ✔️ Triangle

(h) 2, 15, 16
2 + 15 = 17 > 16 ✔
2 + 16 = 18 > 15 ✔
15 + 16 = 31 > 2 ✔ → ✔️ Triangle

✔ Final answers for Problem 1:
(a) No (b) No (c) Yes (d) Yes (e) Yes (f) Yes (g) Yes (h) Yes

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Problem 2: Find range of possible measures for the third side

Rule: If two sides are a and b, then the third side x must satisfy:
|a - b| < x < a + b

(a) 14, 11
|14 - 11| = 3; 14 + 11 = 25 → 3 < x < 25

(b) 6, 10
|6 - 10| = 4; 6 + 10 = 16 → 4 < x < 16

(c) 15, 18
|15 - 18| = 3; 15 + 18 = 33 → 3 < x < 33

(d) 11, 20
|11 - 20| = 9; 11 + 20 = 31 → 9 < x < 31

(e) 47, 21
|47 - 21| = 26; 47 + 21 = 68 → 26 < x < 68

(f) 5, 8
|5 - 8| = 3; 5 + 8 = 13 → 3 < x < 13

(g) 6, 9
|6 - 9| = 3; 6 + 9 = 15 → 3 < x < 15

(h) 17, 19
|17 - 19| = 2; 17 + 19 = 36 → 2 < x < 36

✔ Final answers for Problem 2:
(a) 3 < x < 25 (b) 4 < x < 16 (c) 3 < x < 33 (d) 9 < x < 31
(e) 26 < x < 68 (f) 3 < x < 13 (g) 3 < x < 15 (h) 2 < x < 36

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Problem 3: Name the largest and smallest angle

In any triangle, the largest angle is opposite the longest side, and the smallest angle is opposite the shortest side.

(a) Triangle ABC: AB=8, BC=6, AC=4
Longest side: AB = 8 → opposite angle C → largest angle is ∠C
Shortest side: AC = 4 → opposite angle B → smallest angle is ∠B
Wait — let’s label properly:

Vertices: A, B, C
Sides:
- Opposite A is BC = 6
- Opposite B is AC = 4
- Opposite C is AB = 8

So:
Longest side: AB = 8 → opposite ∠C → largest angle is ∠C
Shortest side: AC = 4 → opposite ∠B → smallest angle is ∠B

But wait — in diagram (a), it shows:
Side opposite A is BC = 6
Side opposite B is AC = 4
Side opposite C is AB = 8
Yes.

So: Largest angle: ∠C; Smallest angle: ∠B

Actually, looking at the diagram again — it’s labeled with sides:
AB = 8, BC = 6, AC = 4
So angles:
∠A is between AB and AC → opposite side BC = 6
∠B is between AB and BC → opposite side AC = 4
∠C is between AC and BC → opposite side AB = 8

So yes:
Largest angle: ∠C (opposite 8)
Smallest angle: ∠B (opposite 4)

(b) Triangle XYZ: XY=16, YZ=9, XZ=8
Longest side: XY = 16 → opposite ∠Z → largest angle is ∠Z
Shortest side: XZ = 8 → opposite ∠Y → smallest angle is ∠Y

Check:
∠X opposite YZ = 9
∠Y opposite XZ = 8
∠Z opposite XY = 16
Yes → Largest: ∠Z; Smallest: ∠Y

(c) Triangle PQR: PQ=5.5, QR=8.5, PR=4.5
Longest side: QR = 8.5 → opposite ∠P → largest angle is ∠P
Shortest side: PR = 4.5 → opposite ∠Q → smallest angle is ∠Q

Check:
∠P opposite QR = 8.5
∠Q opposite PR = 4.5
∠R opposite PQ = 5.5
Yes → Largest: ∠P; Smallest: ∠Q

✔ Final answers for Problem 3:
(a) Largest: ∠C, Smallest: ∠B
(b) Largest: ∠Z, Smallest: ∠Y
(c) Largest: ∠P, Smallest: ∠Q

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Problem 4: List the sides in order, underline the side with the shortest length

Rule: Largest angle → opposite longest side; smallest angle → opposite shortest side.

(a) Triangle ABC: ∠A=60°, ∠B=90°, ∠C=30°
Angles: ∠B (90°) > ∠A (60°) > ∠C (30°)
So sides opposite:
Opposite ∠B is AC → longest
Opposite ∠A is BC → middle
Opposite ∠C is AB → shortest

So sides in order (longest to shortest): AC, BC, AB
Underline AB (shortest)

But the question says “list the sides in order” — probably from shortest to longest? Or just list them with shortest underlined? Let’s assume we list them in increasing order (shortest to longest), and underline the shortest.

Actually, re-read: “List the sides in order, underline the side with the shortest length”

Probably means: write the sides in order (maybe by length), and make sure the shortest is underlined.

Since angles determine side lengths:

Smallest angle: ∠C = 30° → opposite side AB → shortest
Middle angle: ∠A = 60° → opposite side BC
Largest angle: ∠B = 90° → opposite side AC → longest

So sides in order (shortest to longest): AB, BC, AC
Underline AB

(b) Triangle PQR: ∠P=125°, ∠Q=30°, ∠R=25°
Angles: ∠P (125°) > ∠Q (30°) > ∠R (25°)
Opposite sides:
∠P → QR → longest
∠Q → PR → middle
∠R → PQ → shortest

So sides in order (shortest to longest): PQ, PR, QR
Underline PQ

(c) Triangle STU: ∠S=80°, ∠T=60°, ∠U=40°
Angles: ∠S (80°) > ∠T (60°) > ∠U (40°)
Opposite sides:
∠S → TU → longest
∠T → SU → middle
∠U → ST → shortest

So sides in order (shortest to longest): ST, SU, TU
Underline ST

✔ Final answers for Problem 4:
(a) AB, BC, AC — underline AB
(b) PQ, PR, QR — underline PQ
(c) ST, SU, TU — underline ST

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Final Answer:
1. (a) No (b) No (c) Yes (d) Yes (e) Yes (f) Yes (g) Yes (h) Yes
2. (a) 3 < x < 25 (b) 4 < x < 16 (c) 3 < x < 33 (d) 9 < x < 31 (e) 26 < x < 68 (f) 3 < x < 13 (g) 3 < x < 15 (h) 2 < x < 36
3. (a) Largest: ∠C, Smallest: ∠B (b) Largest: ∠Z, Smallest: ∠Y (c) Largest: ∠P, Smallest: ∠Q
4. (a) AB, BC, AC — underline AB (b) PQ, PR, QR — underline PQ (c) ST, SU, TU — underline ST