1. **State the problem:** Given two side lengths of a triangle, determine which of the given options can be the third side length. 2. **Recall the triangle inequality theorem:** For any triangle with sides $a$, $b$, and $c$, the sum of any two sides must be greater than the third side: $$a + b > c, \quad a + c > b, \quad b + c > a$$ 3. **First problem:** Given sides 11 mm and 13 mm, check each option for the third side $x$: - The third side must satisfy: $$|11 - 13| < x < 11 + 13$$ $$2 < x < 24$$ 4. **Check each option:** - a) 4 mm: $2 < 4 < 24$ ✔️ possible - b) 5 mm: $2 < 5 < 24$ ✔️ possible - c) 7 mm: $2 < 7 < 24$ ✔️ possible - d) 32 mm: $2 < 32 < 24$ ❌ not possible 5. **Second problem:** Given sides 3 in and 6 in, check each option for the third side $y$: - The third side must satisfy: $$|3 - 6| < y < 3 + 6$$ $$3 < y < 9$$ 6. **Check each option:** - a) 4 in: $3 < 4 < 9$ ✔️ possible - b) 5 in: $3 < 5 < 9$ ✔️ possible - c) 7 in: $3 < 7 < 9$ ✔️ possible - d) 32 in: $3 < 32 < 9$ ❌ not possible **Final answers:** - For 11 mm and 13 mm: options a), b), c) are possible. - For 3 in and 6 in: options a), b), c) are possible.