1. **Problem:** Determine the number of triangles that can be created given $\alpha$, $a$, and $b$. We use the Law of Sines: $$\frac{a}{\sin \alpha} = \frac{b}{\sin \beta}$$ The number of triangles depends on the value of $a$ relative to $b \sin \alpha$ and $b$. --- **1.** $\alpha=110^\circ$, $a=81$, $b=99$ Calculate $b \sin \alpha = 99 \times \sin 110^\circ$. $$\sin 110^\circ = \sin (180^\circ - 70^\circ) = \sin 70^\circ \approx 0.9397$$ $$b \sin \alpha = 99 \times 0.9397 = 93.03$$ Since $a=81 < 93.03$, and $a < b$, there is **1 triangle**. --- **2.** $\alpha=66^\circ$, $a=73$, $b=71$ Calculate $b \sin \alpha = 71 \times \sin 66^\circ$. $$\sin 66^\circ \approx 0.9135$$ $$b \sin \alpha = 71 \times 0.9135 = 64.06$$ Since $a=73 > 64.06$ and $a > b$, there is **1 triangle**. --- **3.** $\alpha=135^\circ$, $a=45$, $b=50$ Calculate $b \sin \alpha = 50 \times \sin 135^\circ$. $$\sin 135^\circ = \sin 45^\circ = 0.7071$$ $$b \sin \alpha = 50 \times 0.7071 = 35.36$$ Since $a=45 > 35.36$ and $a < b$, there is **1 triangle**. --- **4.** $\alpha=60^\circ$, $a=15\sqrt{3}$, $b=30$ Calculate $a$ numerically: $$15\sqrt{3} = 15 \times 1.732 = 25.98$$ Calculate $b \sin \alpha = 30 \times \sin 60^\circ$. $$\sin 60^\circ = 0.8660$$ $$b \sin \alpha = 30 \times 0.8660 = 25.98$$ Since $a = b \sin \alpha$, there is **1 right triangle**. --- **Final answers for I:** 1. 1 triangle 2. 1 triangle 3. 1 triangle 4. 1 triangle --- **Slug:** triangle count **Subject:** trigonometry