Which sets of measurements could be the side lengths of a triangle? Select each correct answer. A \( 4 \mathrm{~cm}, 8 \mathrm{~cm}, 13 \mathrm{~cm} \) B \( 3 \mathrm{~cm}, 3 \mathrm{~cm}, 3 \mathrm{~cm} \) C \( 6 \mathrm{~cm}, 7 \mathrm{~cm}, 8 \mathrm{~cm} \) D a \( 7 \mathrm{~cm}, 7 \mathrm{~cm}, 10 \mathrm{~cm} \) E \( 5 \mathrm{~cm}, 9 \mathrm{~cm}, 14 \mathrm{~cm} \)

To determine which sets of measurements can be the side lengths of a triangle, we use the **Triangle Inequality Theorem**. This theorem states that for any triangle, the sum of the lengths of any two sides must be **greater** than the length of the remaining side. Let's evaluate each option: - **A. \(4 \text{ cm}, 8 \text{ cm}, 13 \text{ cm}\)** - \(4 + 8 = 12 \), and \(12\) is **not greater** than \(13\). - **Not a valid triangle.** - **B. \(3 \text{ cm}, 3 \text{ cm}, 3 \text{ cm}\)** - \(3 + 3 = 6 > 3\) - All combinations satisfy the inequality. - **Valid triangle.** - **C. \(6 \text{ cm}, 7 \text{ cm}, 8 \text{ cm}\)** - \(6 + 7 = 13 > 8\) - \(6 + 8 = 14 > 7\) - \(7 + 8 = 15 > 6\) - **Valid triangle.** - **D. \(7 \text{ cm}, 7 \text{ cm}, 10 \text{ cm}\)** - \(7 + 7 = 14 > 10\) - \(7 + 10 = 17 > 7\) - \(7 + 10 = 17 > 7\) - **Valid triangle.** - **E. \(5 \text{ cm}, 9 \text{ cm}, 14 \text{ cm}\)** - \(5 + 9 = 14\), and \(14\) is **not greater** than \(14\). - **Not a valid triangle.** **Therefore, the sets of measurements that could be the side lengths of a triangle are:** **B, C, and D**