Triangle Smallest Angle 898634

1. **State the problem:** We are given a triangle with angles labeled as $ (2x+12)^\circ $, $ 56^\circ $, and $ (9x-20)^\circ $. We need to find the measure of the smallest angle. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of any triangle is always $180^\circ$. So, $$ (2x+12) + 56 + (9x-20) = 180 $$ 3. **Simplify the equation:** Combine like terms: $$ 2x + 12 + 56 + 9x - 20 = 180 $$ $$ (2x + 9x) + (12 + 56 - 20) = 180 $$ $$ 11x + 48 = 180 $$ 4. **Isolate $x$:** Subtract 48 from both sides: $$ 11x + \cancel{48} - \cancel{48} = 180 - 48 $$ $$ 11x = 132 $$ 5. **Solve for $x$:** Divide both sides by 11: $$ \frac{11x}{\cancel{11}} = \frac{132}{\cancel{11}} $$ $$ x = 12 $$ 6. **Find each angle measure:** Substitute $x=12$ into each angle expression: - Left vertex angle: $2x + 12 = 2(12) + 12 = 24 + 12 = 36^\circ$ - Top-right vertex angle: $56^\circ$ (given) - Bottom vertex angle: $9x - 20 = 9(12) - 20 = 108 - 20 = 88^\circ$ 7. **Identify the smallest angle:** Among $36^\circ$, $56^\circ$, and $88^\circ$, the smallest angle is $36^\circ$. **Final answer:** The measure of the smallest angle is $\boxed{36^\circ}$.