All changes saved 2. Which of the following sets of measurements can form exactly 1 triangle? \( a=11, b=15, A=36^{\circ} \) \( a=12, b=27, A=37^{\circ} \) \( a=7, b=5, A=19^{\circ} \) \( a=6, b=10, A=42^{\circ} \) \( a \)

To determine which set of measurements can form **exactly one triangle**, we'll analyze each option using the **Side-Side-Angle (SSA)** condition. The SSA scenario can sometimes lead to: 1. **No triangle** 2. **Exactly one triangle** 3. **Two distinct triangles** ### Key Concepts: - **Height Calculation**: \( h = b \cdot \sin(A) \) - **Triangle Conditions**: - If \( a < h \): **No triangle** - If \( a = h \): **One right triangle** - If \( h < a < b \) and \( A \) is **acute**: **Two triangles** - If \( a \geq b \): **One triangle** ### Analyzing Each Option: 1. **\( a = 11 \), \( b = 15 \), \( A = 36^\circ \)** - \( h = 15 \cdot \sin(36^\circ) \approx 8.817 \) - \( a = 11 > h \) and \( a < b \) - **Result**: **Two triangles** 2. **\( a = 12 \), \( b = 27 \), \( A = 37^\circ \)** - \( h = 27 \cdot \sin(37^\circ) \approx 16.25 \) - \( a = 12 < h \) - **Result**: **No triangle** 3. **\( a = 7 \), \( b = 5 \), \( A = 19^\circ \)** - \( h = 5 \cdot \sin(19^\circ) \approx 1.625 \) - \( a = 7 > h \) and \( a > b \) - **Result**: **One triangle** 4. **\( a = 6 \), \( b = 10 \), \( A = 42^\circ \)** - \( h = 10 \cdot \sin(42^\circ) \approx 6.691 \) - \( a = 6 < h \) - **Result**: **No triangle** ### Conclusion: Among the provided options, **only the third set** (\( a = 7 \), \( b = 5 \), \( A = 19^\circ \)) **forms exactly one triangle**. **Answer:** Only the measurements \( a=7 \), \( b=5 \), and \( A=19^{\circ} \) form exactly one triangle.