Answer
Option (D)
Work Step by Step
#### **A. $A = 50^\circ, B = 50^\circ, C = 80^\circ$**
* All **three angles** are given.
* **Not enough**: angle-angle-angle (**AAA**) only determines **shape**, not **size**.
* 🔴 **Does not determine a unique triangle.**
---
#### **B. $a = 3, b = 5, c = 20$**
* This is **SSS**, but check the triangle inequality:
$$
a + b = 3 + 5 = 8 < 20 = c
$$
* Violates triangle inequality → no triangle possible.
* 🔴 **Does not determine a triangle at all.**
---
#### **C. $A = 40^\circ, B = 20^\circ, C = 30^\circ$**
* Three angles again.
* $$
A + B + C = 90^\circ \neq 180^\circ
$$
* Invalid triangle (angle sum incorrect).
* 🔴 **Does not determine a triangle.**
---
#### **D. $a = 7, b = 24, c = 25$**
* Three sides given → **SSS**
* Check triangle inequality:
$$
a + b = 7 + 24 = 31 > 25 \quad \text{✓} \\
a + c = 7 + 25 = 32 > 24 \quad \text{✓} \\
b + c = 24 + 25 = 49 > 7 \quad \text{✓}
$$
* All inequalities satisfied → triangle exists.
* ✅ SSS → **Unique triangle**
---
### **Final Answer**
$$
\boxed{\text{D. } a = 7, b = 24, c = 25}
$$
