Triangle Inequality of Sides worksheet with problems to order triangle sides and identify largest and smallest sides based on angle measures.

To solve these problems, we use the Triangle Inequality of Sides rule:

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

So, to order sides from largest to smallest (or find largest/smallest side), we just look at the angles — bigger angle → longer opposite side.

Let’s go step by step for each problem.

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Problem 1: Triangle AYG

Angles:
- ∠A = 86°
- ∠Y = 51°
- ∠G = 43°

Order angles from largest to smallest:
86° > 51° > 43° → so sides opposite them:
Side opposite ∠A is YG
Side opposite ∠Y is AG
Side opposite ∠G is AY

So sides from largest to smallest:
YG, AG, AY

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Problem 2: Triangle TNH

Angles:
- ∠T = 68°
- ∠N = 68°
- ∠H = 44°

Wait — two angles are equal? That means it’s an isosceles triangle.

Angles: 68°, 68°, 44° → largest angles are both 68°, smallest is 44°.

Sides opposite:
- Opposite ∠T (68°) → NH
- Opposite ∠N (68°) → TH
- Opposite ∠H (44°) → TN

Since ∠T = ∠N, then sides NH = TH (equal length). But the question says “order from largest to smallest”. Since two are equal, we can list them together or in either order.

But typically, if two are equal, we still write them as tied for largest.

So: largest sides: NH and TH (tie), smallest: TN

But since the blank probably expects a sequence, and they’re equal, we can write:

NH, TH, TN OR TH, NH, TN — but let’s check vertex labels.

Triangle is labeled T, N, H.

Actually, looking at diagram: points are T, N, H — with angles given at each.

Angle at T = 68°, at N = 68°, at H = 44°.

So side opposite T is NH
Side opposite N is TH
Side opposite H is TN

So yes: NH and TH are equal and largest, TN smallest.

We’ll write: NH, TH, TN (since NH and TH are same size, order between them doesn’t matter)

But maybe the worksheet expects strict ordering — but mathematically, they’re equal. We’ll note that.

However, for consistency, we'll list the two equal ones first, then the smallest.

Final order: NH, TH, TN

*(Note: Some might prefer writing TH, NH, TN — same thing)*

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Problem 3: ΔTYF

Given:
- m∠T = 43°
- m∠F = 81°
- m∠Y = 56°

Check sum: 43 + 81 + 56 = 180° → good.

Order angles: 81° > 56° > 43°

Sides opposite:
- Opposite ∠F (81°) → TY
- Opposite ∠Y (56°) → TF
- Opposite ∠T (43°) → FY

So sides from largest to smallest:
TY, TF, FY

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Problem 4: ΔYGT

Given:
- m∠G = 60°
- m∠Y = 57°
- m∠T = 63°

Sum: 60+57+63=180 → good.

Order angles: 63° > 60° > 57°

Sides opposite:
- Opposite ∠T (63°) → YG
- Opposite ∠G (60°) → YT
- Opposite ∠Y (57°) → GT

So sides from largest to smallest:
YG, YT, GT

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Problem 5: Triangle HVI

Angles:
- ∠H = 87°
- ∠V = 46°
- ∠I = 47°

Largest angle: 87° → opposite side VI
Smallest angle: 46° → opposite side HI

So:
largest: VI
smallest: HI

*(Note: Side opposite H is VI; opposite V is HI; opposite I is HV)*

Yes:
∠H=87° → opposite VI
∠V=46° → opposite HI
∠I=47° → opposite HV

So largest side: VI (opposite 87°)
Smallest side: HI (opposite 46°)

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Problem 6: Triangle MBZ

Angles:
- ∠M = 88°
- ∠B = 44°
- ∠Z = 48°

Largest angle: 88° → opposite BZ
Smallest angle: 44° → opposite MZ

So:
largest: BZ
smallest: MZ

*(Side opposite M is BZ; opposite B is MZ; opposite Z is MB)*

Yes.

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Problem 7: ΔCPS

Given:
- m∠S = 67°
- m∠P = 59°
- m∠C = 54°

Largest angle: 67° → opposite CP
Smallest angle: 54° → opposite PS

So:
largest: CP
smallest: PS

*(Side opposite S is CP; opposite P is CS; opposite C is PS)*

Yes.

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Problem 8: ΔZHT

Given:
- m∠T = 69°
- m∠H = 58°
- m∠Z = 53°

Largest angle: 69° → opposite ZH
Smallest angle: 53° → opposite HT

So:
largest: ZH
smallest: HT

*(Side opposite T is ZH; opposite H is ZT; opposite Z is HT)*

Yes.

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Now, compiling all answers clearly.

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Final Answer:

1) YG, AG, AY
2) NH, TH, TN
3) TY, TF, FY
4) YG, YT, GT
5) largest: VI, smallest: HI
6) largest: BZ, smallest: MZ
7) largest: CP, smallest: PS
8) largest: ZH, smallest: HT