To solveTo this determine problem which, set we apply of the triangle side inequality theorem, which lengths states:

For anycannot form triangle a with sides triangle, we $a use$, the ** $btriangle$, inequality theorem and**. (c The\ theorem) states (:

where ( Thea sum \ of theleq b lengths of \ anyleq two sides of a triangle c\ must be greater)): 1 than the length of the third side.

###. Let's evaluate each option (:

####a A. $4, 6, +8$

• b $4 + 6 = > c\ 10 > 8$ ✔
• (4 + 8), = 212 > 6. $$$$ ✔
• $a6 + 8 = +14 > c4$ ✔

• b This can form a triangle.

#### B. (10, 9, ), 2\3. $b + c > a$.

Let) ’s- evaluate ( each option10:

+### Option A : $49, 6, =8$

• $4 + 6 = 10 > 8$ ✔️
• $4 + 8 = 12 > 6$ ✔️
• $6 + 8 = 14 > 4$ ✔️ Valid triangle

Option19 > B: $10 2$ ✔

• $10, + 2 = 12 > 9$ ✔ 9, 2)
• 10 + - \(9 + 2 = 11 > 10 ✔
• This can form a triangle.

C. $15, 8, 6$

• $15 + 8 = 23 > 6$ ✔
• $15 + 6 = 21 > 8$ ✔
• $8 + 6 = 14 < 15$ ✘
• This cannot form a triangle because the sum of the two smaller sides ($8 + 6 = 14$) is not greater than the largest side ($15$).

D. $18, 10, 9$

• $18 + 910 = = 28 > 9$ ✔
• $18 + 9 = 27 > 10$ ✔
• $10 + 9 = 19 > 18$ ✔
• This can form a triangle.

Answer:

The correct19 > 2) ✔️

• $10 + answer2 = 12 > 9$ ✔ is **C. (15️,
• $9 + 2 = 11 > 10$ ✔️ Valid triangle

Option C: $15, 88,, 6$

• $15 + 8 =6 \23 > 6$ ✔️)**
• (15 + because it violates the6 = triangle inequality21 theorem >.

8Do\ you) ✔ have️ any questions or- would ( you8 like + me to6 = explain anything 14 further?

\Herenot are> 515 related\ questions) to consider:

1. ❌ What ** isNot a valid triangle**

the### triangle Option inequality D theorem: and ( why18 is, it essential10? ,2 .9 Can\ three)

• equal ( side18 lengths + always form10 a = triangle 28? 3 >. What9) ✔️ are the- types of triangles based ( on18 side + length ? 94 =. How27 does > the 10 triangle inequality\ relate) to ✔ real️-world geometry- ( problems10? +5 .9 How = do 19 angles and sides influence > the 18\ validity) ✔ of️ a triangle**?

Valid** triangleTip**

:### Always Conclusion verify: theThe triangle answer inequality is for ** allC three combinations: 15 of cm sides when, 8 cm testing for a valid, triangle! 6 cm, as it does not satisfy the triangle inequality theorem.

Let me know if you want further clarification or additional examples.

Here are 5 related questions to practice:

1. What is the triangle inequality theorem, and why is it necessary for defining triangles?
2. Can a triangle have side lengths $5, 5, 10$?
3. What are the conditions for forming a right triangle given three sides?
4. How does the triangle inequality theorem extend to quadrilaterals or polygons?
5. Can three sides of a triangle be irrational numbers?

Tip: When checking if three sides form a triangle, always start by summing the two smallest sides to see if they are greater than the largest.