Geometry worksheet with triangle problems for students to solve.

Let’s go through the problems step by step.

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Part 1: Decide whether each set of numbers can be the side lengths of 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

But we only need to check the smallest two sides added together — if their sum is greater than the largest side, then it works.

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Problem 1: 4, 6, 12
Smallest two: 4 + 6 = 10 → Is 10 > 12? ✘ No → Not a triangle.

Problem 2: 2, 8, 10.7
2 + 8 = 10 → Is 10 > 10.7? ✘ No → Not a triangle.

Problem 3: 1.5, 3.4, and 3.7
1.5 + 3.4 = 4.9 → Is 4.9 > 3.7? ✔ Yes → Triangle!

Problem 4: 0.1, 0.1, and 0.1
0.1 + 0.1 = 0.2 → Is 0.2 > 0.1? ✔ Yes → Triangle! (Equilateral)

Problem 5: 9, 10, and 11
9 + 10 = 19 → Is 19 > 11? ✔ Yes → Triangle!

Problem 6: 3.25, 2.1, and 12.5
3.25 + 2.1 = 5.35 → Is 5.35 > 12.5? ✘ No → Not a triangle.

Problem 7: 400, 200, and 300
200 + 300 = 500 → Is 500 > 400? ✔ Yes → Triangle!

Problem 8: 100, 200, and 300
100 + 200 = 300 → Is 300 > 300? ✘ No (must be *greater*, not equal) → Not a triangle.

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Part 2: Given two sides, find possible values for the third side.

Use the rule:
> Third side must be greater than the difference of the two sides and less than the sum.

So if sides are a and b, then:
|a - b| < third side < a + b

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Problem 9: 18 and 14
Difference: 18 - 14 = 4
Sum: 18 + 14 = 32
→ Third side x: 4 < x < 32

Problem 10: 5 and 13
13 - 5 = 8
13 + 5 = 18
→ 8 < x < 18

Problem 11: 124 and 13
124 - 13 = 111
124 + 13 = 137
→ 111 < x < 137

Problem 12: 2.7 and 1.9
2.7 - 1.9 = 0.8
2.7 + 1.9 = 4.6
→ 0.8 < x < 4.6

Problem 13: 44 and 11
44 - 11 = 33
44 + 11 = 55
→ 33 < x < 55

Problem 14: 7 and 12
12 - 7 = 5
12 + 7 = 19
→ 5 < x < 19

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Part 3: Put angles in order from smallest to largest.

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

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Triangle 15: Sides 11, 12, 16
Shortest side: 11 → smallest angle opposite it → ∠C (since side AB=16 is opposite C, wait — let’s label properly.)

Actually, looking at diagram:

- Side opposite ∠A is BC = 12
- Side opposite ∠B is AC = 16
- Side opposite ∠C is AB = 11

So:

- Shortest side: AB = 11 → opposite ∠C → smallest angle
- Middle side: BC = 12 → opposite ∠A
- Longest side: AC = 16 → opposite ∠B → largest angle

Order: ∠C, ∠A, ∠B

Answer: ∠C, ∠A, ∠B

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Triangle 16: Sides 13, 20, 22

Labeling from diagram:

- Side opposite ∠X is YZ = 20
- Side opposite ∠Y is XZ = 22
- Side opposite ∠Z is XY = 13

So:

- Shortest side: XY = 13 → opposite ∠Z → smallest
- Middle: YZ = 20 → opposite ∠X
- Longest: XZ = 22 → opposite ∠Y → largest

Order: ∠Z, ∠X, ∠Y

Answer: ∠Z, ∠X, ∠Y

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Part 4: Put sides in order from shortest to longest.

Again, shortest side is opposite smallest angle, etc.

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Triangle 17: Angles 72°, 61°, 47°

Angles: 47° < 61° < 72°

Sides opposite:

- 47° → opposite side v
- 61° → opposite side u
- 72° → opposite side w

So sides: v (opposite 47°), u (opposite 61°), w (opposite 72°)

Order: v, u, w

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Triangle 18: Angles 50°, 63°, 67°

Angles: 50° < 63° < 67°

Sides opposite:

- 50° → opposite side x
- 63° → opposite side z
- 67° → opposite side y

So sides: x, z, y

Order: x, z, y

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Final Answer:
1. No
2. No
3. Yes
4. Yes
5. Yes
6. No
7. Yes
8. No
9. 4 < x < 32
10. 8 < x < 18
11. 111 < x < 137
12. 0.8 < x < 4.6
13. 33 < x < 55
14. 5 < x < 19
15. ∠C, ∠A, ∠B
16. ∠Z, ∠X, ∠Y
17. v, u, w
18. x, z, y