Geometry Worksheets | Triangle Worksheets - Free Printable

Let's solve each problem step by step using the Triangle Inequality of Angles, which states:

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

We will use this principle to order angles from largest to smallest and identify the largest and smallest angles.

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1) Triangle IMU

- Sides:
- IM = 23 yd
- IU = 27 yd
- MU = 10 yd

Order sides from longest to shortest:
- IU (27) > IM (23) > MU (10)

So, angles opposite:
- ∠M (opposite IU = 27) → largest
- ∠U (opposite IM = 23) → middle
- ∠I (opposite MU = 10) → smallest

✔ Answer: ∠M, ∠U, ∠I

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2) Triangle PQY

- Sides:
- PQ = 17 in
- PY = 21 in
- QY = 12 in

Order sides:
- PY (21) > PQ (17) > QY (12)

Angles opposite:
- ∠Q (opposite PY) → largest
- ∠Y (opposite PQ) → middle
- ∠P (opposite QY) → smallest

✔ Answer: ∠Q, ∠Y, ∠P

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3) Triangle IHA

Given:
- HA = 12 mi
- IH = 32 mi
- AI = 22 mi

Sides:
- IH = 32 (longest)
- AI = 22 (middle)
- HA = 12 (shortest)

Angles opposite:
- ∠A (opposite IH) → largest
- ∠H (opposite AI) → middle
- ∠I (opposite HA) → smallest

✔ Answer: ∠A, ∠H, ∠I

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4) Triangle VBN

Given:
- BN = 15 ft
- VB = 22 ft
- NV = 21 ft

Sides:
- VB = 22 (longest)
- NV = 21 (middle)
- BN = 15 (shortest)

Angles opposite:
- ∠N (opposite VB) → largest
- ∠B (opposite NV) → middle
- ∠V (opposite BN) → smallest

✔ Answer: ∠N, ∠B, ∠V

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5) Triangle ADC

- AD = 18 in
- AC = 30 in
- DC = 15 in

Sides:
- AC = 30 (longest)
- AD = 18 (middle)
- DC = 15 (shortest)

Angles opposite:
- ∠D (opposite AC) → largest
- ∠C (opposite AD) → middle
- ∠A (opposite DC) → smallest

✔ Largest: ∠D
✔ Smallest: ∠A

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6) Triangle VJS

- VJ = 20 yd
- VS = 37 yd
- JS = 21 yd

Sides:
- VS = 37 (longest)
- JS = 21 (middle)
- VJ = 20 (shortest)

Angles opposite:
- ∠J (opposite VS) → largest
- ∠V (opposite JS) → middle
- ∠S (opposite VJ) → smallest

✔ Largest: ∠J
✔ Smallest: ∠S

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7) Triangle OGI

Given:
- OG = 21 in
- IO = 14 in
- GI = 10 in

Note: Triangle is OGI, so vertices are O, G, I.

Sides:
- OG = 21 (longest)
- IO = 14 (middle)
- GI = 10 (shortest)

Angles opposite:
- ∠I (opposite OG) → largest
- ∠G (opposite IO) → middle
- ∠O (opposite GI) → smallest

✔ Largest: ∠I
✔ Smallest: ∠O

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8) Triangle QSP

Given:
- SP = 8 m
- QS = 15 m
- PQ = 14 m

Sides:
- QS = 15 (longest)
- PQ = 14 (middle)
- SP = 8 (shortest)

Angles opposite:
- ∠P (opposite QS) → largest
- ∠Q (opposite SP) → middle
- ∠S (opposite PQ) → smallest

Wait — let’s be careful:

In triangle QSP:
- Side opposite ∠Q is SP = 8
- Side opposite ∠S is PQ = 14
- Side opposite ∠P is QS = 15

So:
- Largest side: QS = 15 → opposite ∠P → largest angle
- Middle side: PQ = 14 → opposite ∠S → middle
- Shortest side: SP = 8 → opposite ∠Q → smallest

✔ Largest: ∠P
✔ Smallest: ∠Q

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✔ Final Answers:

#### Order angles from largest to smallest:
1) ∠M, ∠U, ∠I
2) ∠Q, ∠Y, ∠P
3) ∠A, ∠H, ∠I
4) ∠N, ∠B, ∠V

#### Name largest and smallest angle:
5) Largest: ∠D, Smallest: ∠A
6) Largest: ∠J, Smallest: ∠S
7) Largest: ∠I, Smallest: ∠O
8) Largest: ∠P, Smallest: ∠Q

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