Triangle Inequality Theorem Worksheet for math practice, featuring problems on triangle side lengths and angles.
Triangle Inequality Theorem Worksheet with four sections: determining if sets of numbers can form a triangle, finding the range of possible measures for the third side, naming the largest and smallest angles, and listing sides in order with the shortest length underlined.
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Step-by-step solution for: Triangle Inequality Theorem Worksheets - Math Monks
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Show Answer Key & Explanations
Step-by-step solution for: Triangle Inequality Theorem Worksheets - Math Monks
Let’s solve 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.
All three combinations must work.
Let’s check each:
(a) 5, 10, 15
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 ✔
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 ✔
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 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 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
10 - 6 = 4; 6 + 10 = 16 → 4 < x < 16
(c) 15, 18
18 - 15 = 3; 15 + 18 = 33 → 3 < x < 33
(d) 11, 20
20 - 11 = 9; 11 + 20 = 31 → 9 < x < 31
(e) 47, 21
47 - 21 = 26; 47 + 21 = 68 → 26 < x < 68
(f) 5, 8
8 - 5 = 3; 5 + 8 = 13 → 3 < x < 13
(g) 6, 9
9 - 6 = 3; 6 + 9 = 15 → 3 < x < 15
(h) 17, 19
19 - 17 = 2; 17 + 19 = 36 → 2 < x < 36
✔ Final 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 a triangle, largest angle is opposite the longest side, 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:
Points: A, B, C
Sides:
AB = 8 → opposite angle C
BC = 6 → opposite angle A
AC = 4 → opposite angle B
So:
Longest side AB → opposite ∠C → largest angle: ∠C
Shortest side AC → opposite ∠B → smallest angle: ∠B
✔ (a) Largest: ∠C, Smallest: ∠B
(b) Triangle XYZ: XY=16, YZ=9, ZX=8
Longest side: XY=16 → opposite angle Z → largest angle: ∠Z
Shortest side: ZX=8 → opposite angle Y → smallest angle: ∠Y
✔ (b) Largest: ∠Z, Smallest: ∠Y
(c) Triangle PQR: PQ=5.5, QR=8.5, RP=4.5
Longest side: QR=8.5 → opposite angle P → largest angle: ∠P
Shortest side: RP=4.5 → opposite angle Q → smallest angle: ∠Q
✔ (c) Largest: ∠P, Smallest: ∠Q
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Problem 4: List sides in order, underline shortest side
In a triangle, side opposite larger angle is longer.
(a) Triangle ABC: ∠A=60°, ∠B=90°, ∠C=30°
Angles: B (90°) > A (60°) > C (30°)
So sides opposite:
Opposite B → AC
Opposite A → BC
Opposite C → AB
So side lengths: AC > BC > AB
Shortest side: AB → underline it
Order: AB, BC, AC → underline AB
✔ (a) AB, BC, AC → underline AB
(b) Triangle PQR: ∠P=125°, ∠Q=30°, ∠R=25°
Angles: P (125°) > Q (30°) > R (25°)
Sides opposite:
Opposite P → QR
Opposite Q → PR
Opposite R → PQ
So: QR > PR > PQ
Shortest side: PQ → underline it
Order: PQ, PR, QR → underline PQ
✔ (b) PQ, PR, QR → underline PQ
(c) Triangle STU: ∠S=80°, ∠T=60°, ∠U=40°
Angles: S (80°) > T (60°) > U (40°)
Sides opposite:
Opposite S → TU
Opposite T → SU
Opposite U → ST
So: TU > SU > ST
Shortest side: ST → underline it
Order: ST, SU, TU → underline ST
✔ (c) ST, SU, TU → underline ST
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Final Answers:
Problem 1:
(a) No (b) No (c) Yes (d) Yes (e) Yes (f) Yes (g) Yes (h) Yes
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
Problem 3:
(a) Largest: ∠C, Smallest: ∠B
(b) Largest: ∠Z, Smallest: ∠Y
(c) Largest: ∠P, Smallest: ∠Q
Problem 4:
(a) AB, BC, AC → underline AB
(b) PQ, PR, QR → underline PQ
(c) ST, SU, TU → underline ST
──────────────────────────────────────
Final Answer:
Problem 1: (a) No, (b) No, (c) Yes, (d) Yes, (e) Yes, (f) Yes, (g) Yes, (h) Yes
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
Problem 3: (a) Largest: ∠C, Smallest: ∠B; (b) Largest: ∠Z, Smallest: ∠Y; (c) Largest: ∠P, Smallest: ∠Q
Problem 4: (a) AB, BC, AC (underline AB); (b) PQ, PR, QR (underline PQ); (c) ST, SU, TU (underline ST)
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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.
All three combinations must work.
Let’s check each:
(a) 5, 10, 15
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 ✔
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 ✔
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 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 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
10 - 6 = 4; 6 + 10 = 16 → 4 < x < 16
(c) 15, 18
18 - 15 = 3; 15 + 18 = 33 → 3 < x < 33
(d) 11, 20
20 - 11 = 9; 11 + 20 = 31 → 9 < x < 31
(e) 47, 21
47 - 21 = 26; 47 + 21 = 68 → 26 < x < 68
(f) 5, 8
8 - 5 = 3; 5 + 8 = 13 → 3 < x < 13
(g) 6, 9
9 - 6 = 3; 6 + 9 = 15 → 3 < x < 15
(h) 17, 19
19 - 17 = 2; 17 + 19 = 36 → 2 < x < 36
✔ Final 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
---
Problem 3: Name the largest and smallest angle
In a triangle, largest angle is opposite the longest side, 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:
Points: A, B, C
Sides:
AB = 8 → opposite angle C
BC = 6 → opposite angle A
AC = 4 → opposite angle B
So:
Longest side AB → opposite ∠C → largest angle: ∠C
Shortest side AC → opposite ∠B → smallest angle: ∠B
✔ (a) Largest: ∠C, Smallest: ∠B
(b) Triangle XYZ: XY=16, YZ=9, ZX=8
Longest side: XY=16 → opposite angle Z → largest angle: ∠Z
Shortest side: ZX=8 → opposite angle Y → smallest angle: ∠Y
✔ (b) Largest: ∠Z, Smallest: ∠Y
(c) Triangle PQR: PQ=5.5, QR=8.5, RP=4.5
Longest side: QR=8.5 → opposite angle P → largest angle: ∠P
Shortest side: RP=4.5 → opposite angle Q → smallest angle: ∠Q
✔ (c) Largest: ∠P, Smallest: ∠Q
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Problem 4: List sides in order, underline shortest side
In a triangle, side opposite larger angle is longer.
(a) Triangle ABC: ∠A=60°, ∠B=90°, ∠C=30°
Angles: B (90°) > A (60°) > C (30°)
So sides opposite:
Opposite B → AC
Opposite A → BC
Opposite C → AB
So side lengths: AC > BC > AB
Shortest side: AB → underline it
Order: AB, BC, AC → underline AB
✔ (a) AB, BC, AC → underline AB
(b) Triangle PQR: ∠P=125°, ∠Q=30°, ∠R=25°
Angles: P (125°) > Q (30°) > R (25°)
Sides opposite:
Opposite P → QR
Opposite Q → PR
Opposite R → PQ
So: QR > PR > PQ
Shortest side: PQ → underline it
Order: PQ, PR, QR → underline PQ
✔ (b) PQ, PR, QR → underline PQ
(c) Triangle STU: ∠S=80°, ∠T=60°, ∠U=40°
Angles: S (80°) > T (60°) > U (40°)
Sides opposite:
Opposite S → TU
Opposite T → SU
Opposite U → ST
So: TU > SU > ST
Shortest side: ST → underline it
Order: ST, SU, TU → underline ST
✔ (c) ST, SU, TU → underline ST
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Final Answers:
Problem 1:
(a) No (b) No (c) Yes (d) Yes (e) Yes (f) Yes (g) Yes (h) Yes
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
Problem 3:
(a) Largest: ∠C, Smallest: ∠B
(b) Largest: ∠Z, Smallest: ∠Y
(c) Largest: ∠P, Smallest: ∠Q
Problem 4:
(a) AB, BC, AC → underline AB
(b) PQ, PR, QR → underline PQ
(c) ST, SU, TU → underline ST
──────────────────────────────────────
Final Answer:
Problem 1: (a) No, (b) No, (c) Yes, (d) Yes, (e) Yes, (f) Yes, (g) Yes, (h) Yes
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
Problem 3: (a) Largest: ∠C, Smallest: ∠B; (b) Largest: ∠Z, Smallest: ∠Y; (c) Largest: ∠P, Smallest: ∠Q
Problem 4: (a) AB, BC, AC (underline AB); (b) PQ, PR, QR (underline PQ); (c) ST, SU, TU (underline ST)
Parent Tip: Review the logic above to help your child master the concept of triangle inequality theorem worksheet pdf.
