Triangle Inequality Theorem Worksheets - Math Monks - Free Printable
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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
Absolutely! Let's solve each problem step by step using the Triangle Inequality Theorem and other triangle properties.
---
## 🔢 Problem 1: Decide whether each set of numbers can form a triangle
> Triangle Inequality Theorem: For any three lengths to form a triangle, the sum of the lengths of *any two sides* must be greater than the length of the remaining side.
>
> That is, for sides $a$, $b$, $c$:
> - $a + b > c$
> - $a + c > b$
> - $b + c > a$
We only need to check if the sum of the two smaller sides is greater than the largest side — if that’s true, the other inequalities automatically hold.
---
- Smallest two: 5 + 10 = 15
- Largest: 15
- Is 15 > 15? ✘ No → Cannot form a triangle
> *Note: This would form a degenerate (flat) triangle — not a valid triangle.*
---
- 6 + 9 = 15
- 15 > 16? ✘ No → Cannot form a triangle
---
- 9 + 40 = 49
- 49 > 41? ✔ Yes → Can form a triangle
---
- 7.5 + 8.5 = 16
- 16 > 14.5? ✔ Yes → Can form a triangle
---
- Sort: 9, 11, 12
- 9 + 11 = 20
- 20 > 12? ✔ Yes → Can form a triangle
---
- 1 + 16 = 17
- 17 > 16? ✔ Yes → Can form a triangle
> *(This is an isosceles triangle)*
---
- Sort: 5, 7, 9
- 5 + 7 = 12
- 12 > 9? ✔ Yes → Can form a triangle
---
- 2 + 15 = 17
- 17 > 16? ✔ Yes → Can form a triangle
---
✔ Final Answers for Problem 1:
| Set | Can Form Triangle? |
|-----|---------------------|
| (a) | ✘ No |
| (b) | ✘ No |
| (c) | ✔ Yes |
| (d) | ✔ Yes |
| (e) | ✔ Yes |
| (f) | ✔ Yes |
| (g) | ✔ Yes |
| (h) | ✔ Yes |
---
## 📏 Problem 2: Find the range of possible measures for the third side
> If two sides are known, say $a$ and $b$, then the third side $x$ must satisfy:
>
> $$ |a - b| < x < a + b $$
This ensures the triangle inequality holds.
---
- Difference: |14 - 11| = 3
- Sum: 14 + 11 = 25
- So, 3 < x < 25
---
- |6 - 10| = 4
- 6 + 10 = 16
- 4 < x < 16
---
- |15 - 18| = 3
- 15 + 18 = 33
- 3 < x < 33
---
- |11 - 20| = 9
- 11 + 20 = 31
- 9 < x < 31
---
- |47 - 21| = 26
- 47 + 21 = 68
- 26 < x < 68
---
- |5 - 8| = 3
- 5 + 8 = 13
- 3 < x < 13
---
- |6 - 9| = 3
- 6 + 9 = 15
- 3 < x < 15
---
- |17 - 19| = 2
- 17 + 19 = 36
- 2 < x < 36
---
✔ Final Answers for Problem 2:
| Pair | Range for Third Side |
|----------|----------------------------|
| (a) 14,11 | 3 < x < 25 |
| (b) 6,10 | 4 < x < 16 |
| (c) 15,18 | 3 < x < 33 |
| (d) 11,20 | 9 < x < 31 |
| (e) 47,21 | 26 < x < 68 |
| (f) 5,8 | 3 < x < 13 |
| (g) 6,9 | 3 < x < 15 |
| (h) 17,19 | 2 < x < 36 |
---
## ⚖️ Problem 3: Name the largest and smallest angle
> In any triangle:
> - The largest angle is opposite the longest side
> - The smallest angle is opposite the shortest side
Let’s analyze each triangle:
---
Sides: AB = 8, BC = 6, AC = 4
- Longest side: AB = 8 → Opposite angle is ∠C
- Shortest side: AC = 4 → Opposite angle is ∠B
✔ Largest angle: ∠C
✔ Smallest angle: ∠B
---
Sides: XY = 16, YZ = 9, XZ = 8
- Longest side: XY = 16 → Opposite angle is ∠Z
- Shortest side: XZ = 8 → Opposite angle is ∠Y
✔ Largest angle: ∠Z
✔ Smallest angle: ∠Y
---
Sides: PQ = 5.5, QR = 8.5, PR = 4.5
- Longest side: QR = 8.5 → Opposite angle is ∠P
- Shortest side: PR = 4.5 → Opposite angle is ∠Q
✔ Largest angle: ∠P
✔ Smallest angle: ∠Q
---
✔ Final Answers for Problem 3:
| Triangle | Largest Angle | Smallest Angle |
|----------|---------------|----------------|
| (a) ABC | ∠C | ∠B |
| (b) XYZ | ∠Z | ∠Y |
| (c) PQR | ∠P | ∠Q |
---
## 📐 Problem 4: List the sides in order, underline the shortest side
> Again, use: Largest angle ↔ Longest side, Smallest angle ↔ Shortest side
---
Angles: ∠A = 60°, ∠B = 90°, ∠C = 30°
- Largest angle: ∠B = 90° → Longest side: AC
- Smallest angle: ∠C = 30° → Shortest side: AB
- Middle angle: ∠A = 60° → Middle side: BC
Order from longest to shortest: AC, BC, AB
✔ Underline shortest: AC, BC, _AB_
---
Angles: ∠P = 125°, ∠Q = 30°, ∠R = 25°
- Largest angle: ∠P = 125° → Longest side: QR
- Smallest angle: ∠R = 25° → Shortest side: PQ
- Middle angle: ∠Q = 30° → Middle side: PR
Order from longest to shortest: QR, PR, PQ
✔ Underline shortest: QR, PR, _PQ_
---
Angles: ∠S = 80°, ∠T = 60°, ∠U = 40°
- Largest angle: ∠S = 80° → Longest side: TU
- Smallest angle: ∠U = 40° → Shortest side: ST
- Middle angle: ∠T = 60° → Middle side: SU
Order from longest to shortest: TU, SU, ST
✔ Underline shortest: TU, SU, _ST_
---
✔ Final Answers for Problem 4:
| Triangle | Sides in Order (Longest to Shortest) | Shortest Side (Underlined) |
|----------|--------------------------------------|----------------------------|
| (a) ABC | AC, BC, AB | AC, BC, _AB_ |
| (b) PQR | QR, PR, PQ | QR, PR, _PQ_ |
| (c) STU | TU, SU, ST | TU, SU, _ST_ |
---
## 🎉 Final Summary
You’ve now solved all 4 problems using the Triangle Inequality Theorem and the relationship between side lengths and opposite angles.
Let me know if you’d like this as a printable answer sheet or with diagrams!
---
---
## 🔢 Problem 1: Decide whether each set of numbers can form a triangle
> Triangle Inequality Theorem: For any three lengths to form a triangle, the sum of the lengths of *any two sides* must be greater than the length of the remaining side.
>
> That is, for sides $a$, $b$, $c$:
> - $a + b > c$
> - $a + c > b$
> - $b + c > a$
We only need to check if the sum of the two smaller sides is greater than the largest side — if that’s true, the other inequalities automatically hold.
---
(a) 5, 10, 15
- Smallest two: 5 + 10 = 15
- Largest: 15
- Is 15 > 15? ✘ No → Cannot form a triangle
> *Note: This would form a degenerate (flat) triangle — not a valid triangle.*
---
(b) 6, 9, 16
- 6 + 9 = 15
- 15 > 16? ✘ No → Cannot form a triangle
---
(c) 9, 40, 41
- 9 + 40 = 49
- 49 > 41? ✔ Yes → Can form a triangle
---
(d) 7.5, 8.5, 14.5
- 7.5 + 8.5 = 16
- 16 > 14.5? ✔ Yes → Can form a triangle
---
(e) 11, 12, 9
- Sort: 9, 11, 12
- 9 + 11 = 20
- 20 > 12? ✔ Yes → Can form a triangle
---
(f) 1, 16, 16
- 1 + 16 = 17
- 17 > 16? ✔ Yes → Can form a triangle
> *(This is an isosceles triangle)*
---
(g) 9, 5, 7
- Sort: 5, 7, 9
- 5 + 7 = 12
- 12 > 9? ✔ Yes → Can form a triangle
---
(h) 2, 15, 16
- 2 + 15 = 17
- 17 > 16? ✔ Yes → Can form a triangle
---
✔ Final Answers for Problem 1:
| Set | Can Form Triangle? |
|-----|---------------------|
| (a) | ✘ No |
| (b) | ✘ No |
| (c) | ✔ Yes |
| (d) | ✔ Yes |
| (e) | ✔ Yes |
| (f) | ✔ Yes |
| (g) | ✔ Yes |
| (h) | ✔ Yes |
---
## 📏 Problem 2: Find the range of possible measures for the third side
> If two sides are known, say $a$ and $b$, then the third side $x$ must satisfy:
>
> $$ |a - b| < x < a + b $$
This ensures the triangle inequality holds.
---
(a) 14, 11
- Difference: |14 - 11| = 3
- Sum: 14 + 11 = 25
- So, 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:
| Pair | Range for Third Side |
|----------|----------------------------|
| (a) 14,11 | 3 < x < 25 |
| (b) 6,10 | 4 < x < 16 |
| (c) 15,18 | 3 < x < 33 |
| (d) 11,20 | 9 < x < 31 |
| (e) 47,21 | 26 < x < 68 |
| (f) 5,8 | 3 < x < 13 |
| (g) 6,9 | 3 < x < 15 |
| (h) 17,19 | 2 < x < 36 |
---
## ⚖️ Problem 3: Name the largest and smallest angle
> In any triangle:
> - The largest angle is opposite the longest side
> - The smallest angle is opposite the shortest side
Let’s analyze each triangle:
---
(a) Triangle ABC
Sides: AB = 8, BC = 6, AC = 4
- Longest side: AB = 8 → Opposite angle is ∠C
- Shortest side: AC = 4 → Opposite angle is ∠B
✔ Largest angle: ∠C
✔ Smallest angle: ∠B
---
(b) Triangle XYZ
Sides: XY = 16, YZ = 9, XZ = 8
- Longest side: XY = 16 → Opposite angle is ∠Z
- Shortest side: XZ = 8 → Opposite angle is ∠Y
✔ Largest angle: ∠Z
✔ Smallest angle: ∠Y
---
(c) Triangle PQR
Sides: PQ = 5.5, QR = 8.5, PR = 4.5
- Longest side: QR = 8.5 → Opposite angle is ∠P
- Shortest side: PR = 4.5 → Opposite angle is ∠Q
✔ Largest angle: ∠P
✔ Smallest angle: ∠Q
---
✔ Final Answers for Problem 3:
| Triangle | Largest Angle | Smallest Angle |
|----------|---------------|----------------|
| (a) ABC | ∠C | ∠B |
| (b) XYZ | ∠Z | ∠Y |
| (c) PQR | ∠P | ∠Q |
---
## 📐 Problem 4: List the sides in order, underline the shortest side
> Again, use: Largest angle ↔ Longest side, Smallest angle ↔ Shortest side
---
(a) Triangle ABC
Angles: ∠A = 60°, ∠B = 90°, ∠C = 30°
- Largest angle: ∠B = 90° → Longest side: AC
- Smallest angle: ∠C = 30° → Shortest side: AB
- Middle angle: ∠A = 60° → Middle side: BC
Order from longest to shortest: AC, BC, AB
✔ Underline shortest: AC, BC, _AB_
---
(b) Triangle PQR
Angles: ∠P = 125°, ∠Q = 30°, ∠R = 25°
- Largest angle: ∠P = 125° → Longest side: QR
- Smallest angle: ∠R = 25° → Shortest side: PQ
- Middle angle: ∠Q = 30° → Middle side: PR
Order from longest to shortest: QR, PR, PQ
✔ Underline shortest: QR, PR, _PQ_
---
(c) Triangle STU
Angles: ∠S = 80°, ∠T = 60°, ∠U = 40°
- Largest angle: ∠S = 80° → Longest side: TU
- Smallest angle: ∠U = 40° → Shortest side: ST
- Middle angle: ∠T = 60° → Middle side: SU
Order from longest to shortest: TU, SU, ST
✔ Underline shortest: TU, SU, _ST_
---
✔ Final Answers for Problem 4:
| Triangle | Sides in Order (Longest to Shortest) | Shortest Side (Underlined) |
|----------|--------------------------------------|----------------------------|
| (a) ABC | AC, BC, AB | AC, BC, _AB_ |
| (b) PQR | QR, PR, PQ | QR, PR, _PQ_ |
| (c) STU | TU, SU, ST | TU, SU, _ST_ |
---
## 🎉 Final Summary
You’ve now solved all 4 problems using the Triangle Inequality Theorem and the relationship between side lengths and opposite angles.
Let me know if you’d like this as a printable answer sheet or with diagrams!
---
Parent Tip: Review the logic above to help your child master the concept of triangle inequality worksheet answers.
