Triangle Inequality Theorem Worksheets - Math Monks - Free Printable

Let’s go step by step through each problem on the worksheet.

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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 three numbers to form a triangle, all three conditions must be true:
- a + b > c
- a + c > b
- b + c > a

But we can simplify: just check if the sum of the two smaller sides is greater than the largest side. If yes → triangle possible. If no → not possible.

(a) 5, 10, 15
Smallest two: 5 + 10 = 15 → NOT greater than 15 → ✘ No

(b) 6, 9, 16
6 + 9 = 15 < 16 → ✘ No

(c) 9, 40, 41
9 + 40 = 49 > 41 → ✔ Yes

(d) 7.5, 8.5, 14.5
7.5 + 8.5 = 16 > 14.5 → ✔ Yes

(e) 11, 12, 9
Sort: 9, 11, 12 → 9 + 11 = 20 > 12 → ✔ Yes

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

(g) 9, 5, 7
Sort: 5, 7, 9 → 5 + 7 = 12 > 9 → ✔ Yes

(h) 2, 15, 16
2 + 15 = 17 > 16 → ✔ Yes

✔ Final answers for #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 third side

Rule: For two given sides *a* and *b*, the third side *x* must satisfy:
|a - b| < x < a + b

That means:
Third side must be greater than the difference and less than the sum of the two sides.

(a) 14, 11
Difference: 14 - 11 = 3
Sum: 14 + 11 = 25
→ Range: 3 < x < 25

(b) 6, 10
Diff: 4, Sum: 16 → 4 < x < 16

(c) 15, 18
Diff: 3, Sum: 33 → 3 < x < 33

(d) 11, 20
Diff: 9, Sum: 31 → 9 < x < 31

(e) 47, 21
Diff: 26, Sum: 68 → 26 < x < 68

(f) 5, 8
Diff: 3, Sum: 13 → 3 < x < 13

(g) 6, 9
Diff: 3, Sum: 15 → 3 < x < 15

(h) 17, 19
Diff: 2, Sum: 36 → 2 < x < 36

✔ Final answers for #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 a triangle, largest angle is opposite the longest side, and smallest angle is opposite the shortest side.

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

Actually, in triangle ABC:
- Side opposite A is BC = 6
- Side opposite B is AC = 4
- Side opposite C is AB = 8

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

Wait — but looking at diagram: it shows right angle at C? Actually, the diagram has labels:

It says:
A —— 8 —— B
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4 6
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C —— 6 —— ? Wait, actually from image description:

From your text:
(a) Triangle with points A, B, C. Sides: AB=8, BC=6, AC=4? But that doesn’t make sense geometrically unless it's labeled differently.

Looking back at user input:
“(a) [triangle] A top, B left, C right. AB=8, BC=6, AC=4” — wait, that would mean:

Actually, standard labeling: side opposite vertex is lowercase, but here they give lengths between vertices.

Assume:
- Side AB = 8
- Side BC = 6
- Side AC = 4

Then angles:
- Angle at A is opposite side BC = 6
- Angle at B is opposite side AC = 4
- Angle at C is opposite side AB = 8

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

But wait — in many diagrams like this, sometimes the right angle is shown. However, since no angle measures are given, we rely only on side lengths.

So for (a):
Largest angle: ∠C
Smallest angle: ∠B

(b) Triangle XYZ: XY=16, YZ=9, XZ=8
Sides: 8 (XZ), 9 (YZ), 16 (XY)

Angles:
- Opposite X: YZ = 9
- Opposite Y: XZ = 8
- Opposite Z: XY = 16

So:
Largest side: XY = 16 → opposite angle Z → ∠Z largest
Smallest side: XZ = 8 → opposite angle Y → ∠Y smallest

(c) Triangle PQR: PQ=5.5, QR=8.5, PR=4.5

Sides: 4.5 (PR), 5.5 (PQ), 8.5 (QR)

Angles:
- Opposite P: QR = 8.5
- Opposite Q: PR = 4.5
- Opposite R: PQ = 5.5

So:
Largest side: QR = 8.5 → opposite angle P → ∠P largest
Smallest side: PR = 4.5 → opposite angle Q → ∠Q smallest

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

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Problem 4: List sides in order, underline shortest

Again, same rule: side opposite larger angle is longer.

Given angles, find which side is opposite which angle.

(a) Triangle ABC: ∠A=60°, ∠B=90°, ∠C=30°

Angles: 30° (∠C), 60° (∠A), 90° (∠B)

Sides opposite:
- Opposite ∠A (60°): side BC
- Opposite ∠B (90°): side AC
- Opposite ∠C (30°): side AB

So side lengths correspond to angle sizes:
Largest angle → longest side → ∠B=90° → side AC
Middle → ∠A=60° → side BC
Smallest → ∠C=30° → side AB

Order from shortest to longest: AB, BC, AC
Underline shortest: AB

(b) Triangle PQR: ∠P=125°, ∠Q=30°, ∠R=25°

Angles: 25° (∠R), 30° (∠Q), 125° (∠P)

Sides opposite:
- Opposite ∠P (125°): QR
- Opposite ∠Q (30°): PR
- Opposite ∠R (25°): PQ

So:
Longest: QR (opposite 125°)
Middle: PR (opposite 30°)
Shortest: PQ (opposite 25°)

Order: PQ, PR, QR
Underline shortest: PQ

(c) Triangle STU: ∠S=80°, ∠T=60°, ∠U=40°

Angles: 40° (∠U), 60° (∠T), 80° (∠S)

Sides opposite:
- Opposite ∠S (80°): TU
- Opposite ∠T (60°): SU
- Opposite ∠U (40°): ST

So:
Longest: TU
Middle: SU
Shortest: ST

Order: ST, SU, TU
Underline shortest: ST

✔ Final answers for #4:
(a) Order: AB, BC, AC → underline AB
(b) Order: PQ, PR, QR → underline PQ
(c) Order: 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)