1. The problem states that two angles in a triangle are 50 degrees and 80 degrees, and we need to find the ratio of the third angle to the sum of the first two angles. 2. Recall that the sum of the interior angles in any triangle is always 180 degrees. This is a fundamental rule in geometry. 3. Let the third angle be $x$ degrees. Then, according to the triangle angle sum property: $$50 + 80 + x = 180$$ 4. Simplify the equation: $$130 + x = 180$$ 5. Solve for $x$: $$x = 180 - 130 = 50$$ 6. Now, find the sum of the first two angles: $$50 + 80 = 130$$ 7. The ratio of the third angle to the sum of the first two angles is: $$\frac{x}{50 + 80} = \frac{50}{130}$$ 8. Simplify the ratio by dividing numerator and denominator by 10: $$\frac{50}{130} = \frac{5}{13}$$ Final answer: The ratio of the third angle to the sum of the first two angles is $\frac{5}{13}$.