Comprehensive worksheet for practicing the Triangle Inequality Theorem, including determining valid side lengths and ordering angles.

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

Rule: For three lengths to form a triangle, the sum of any two sides must be greater than the third side.
We only need to check if the two smaller sides add up to more than the largest side — if that’s true, then all other combinations will also work.

Let’s go one by one:

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

(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 (even though it's very skinny, it works)

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

(h) 2, 15, 16 → 2 + 15 = 17 > 16 → ✔ 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: Two sides given. Find range for third side.

Rule: If two sides are *a* and *b*, then the third side *x* must satisfy:
|a - b| < x < a + b

Let’s apply this:

(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 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

Rule: In a 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
→ Longest side: AB=8 → opposite angle C → Largest angle: ∠C
→ Shortest side: AC=4 → opposite angle B → Smallest angle: ∠B

Wait — let’s label carefully:

Points: A, B, C
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: ∠C
Shortest side = AC = 4 → opposite angle B → Smallest angle: ∠B

✔ (a) Largest: ∠C, Smallest: ∠B

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

Opposite angles:
- Side XY (16) is opposite angle Z → Largest angle: ∠Z
- Side XZ (8) is opposite angle Y → Smallest angle: ∠Y

✔ (b) Largest: ∠Z, Smallest: ∠Y

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

Opposite angles:
- QR (8.5) is opposite angle P → Largest angle: ∠P
- PR (4.5) is opposite angle Q → Smallest angle: ∠Q

✔ (c) Largest: ∠P, Smallest: ∠Q

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Problem 4: List sides in order, underline shortest

Rule: In a triangle, side opposite larger angle is longer. So we list sides from shortest to longest based on opposite angles.

Also, underline the shortest side.

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

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

So side lengths: AB (opposite 30°) < BC (opposite 60°) < AC (opposite 90°)

Order: AB, BC, AC → Underline AB (shortest)

✔ (a) AB, BC, AC → underline AB

(b) Triangle PQR: Angles: ∠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: PQ (opposite 25°) < PR (opposite 30°) < QR (opposite 125°)

Order: PQ, PR, QR → underline PQ

✔ (b) PQ, PR, QR → underline PQ

(c) Triangle STU: Angles: ∠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: ST (opposite 40°) < SU (opposite 60°) < TU (opposite 80°)

Order: ST, SU, TU → underline ST

✔ (c) ST, SU, TU → underline ST

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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