Triangle Inequality Theorem Worksheets - Math Monks - Free Printable

Let's solve each problem on the Triangle Inequality Theorem Worksheet step by step.

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

Use the Triangle Inequality Theorem:
> The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

We check for each set:

(a) 5, 10, 15
- 5 + 10 = 15 → not greater than 15 → ✘ No
- (Equal, not greater — violates inequality)

(b) 6, 9, 16
- 6 + 9 = 15 < 16 → ✘ No

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

(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 ✔ → ✔️ Yes

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

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

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

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

✔ 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: Two sides of a triangle have the following measures. Find the range of possible measures for the third side.

Let the two given sides be $ 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:
- Sides: AB = 8, BC = 6, AC = 4
- So, longest side = AB = 8 → opposite angle ∠C
- shortest side = AC = 4 → opposite angle ∠B

→ Largest angle: ∠C
→ Smallest angle: ∠B

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

→ Largest angle: ∠Z
→ Smallest angle: ∠Y

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

→ Largest angle: ∠P
→ Smallest angle: ∠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

Use: Smaller angle → opposite shorter side

(a) Triangle ABC:
- Angles: ∠A = 60°, ∠B = 90°, ∠C = 30°
- Order of angles: ∠C (30°) < ∠A (60°) < ∠B (90°)
- So sides opposite:
- ∠C → AB
- ∠A → BC
- ∠B → AC

→ Side lengths: AB < BC < AC
→ Shortest side: AB

List: AB, BC, AC → underline AB

(b) Triangle PQR:
- ∠P = 125°, ∠Q = 30°, ∠R = 25°
- Order of angles: ∠R (25°) < ∠Q (30°) < ∠P (125°)
- Opposite sides:
- ∠R → PQ
- ∠Q → PR
- ∠P → QR

→ Sides: PQ < PR < QR
→ Shortest side: PQ

List: PQ, PR, QR → underline PQ

(c) Triangle STU:
- ∠S = 80°, ∠T = 60°, ∠U = 40°
- Order: ∠U (40°) < ∠T (60°) < ∠S (80°)
- Opposite sides:
- ∠U → ST
- ∠T → SU
- ∠S → TU

→ Sides: ST < SU < TU
→ Shortest side: ST

List: ST, SU, TU → underline ST

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

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✔ Final Summary of All Answers:

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#### Problem 1: Can form a triangle?
(a) No
(b) No
(c) Yes
(d) Yes
(e) Yes
(f) Yes
(g) Yes
(h) Yes

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#### Problem 2: Range of third side
(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: Largest and smallest angle
(a) Largest: ∠C, Smallest: ∠B
(b) Largest: ∠Z, Smallest: ∠Y
(c) Largest: ∠P, Smallest: ∠Q

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#### Problem 4: List sides, underline shortest
(a) AB, BC, AC → AB
(b) PQ, PR, QR → PQ
(c) ST, SU, TU → ST

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