1. The ratio of the measures of the three angles in a triangle is \( 14: 5: 11 \). Find the measures of the angles. Show all work.

To find the measures of the angles in the triangle with a ratio of \( 14: 5: 11 \), we start by letting the measures of the angles be \( 14x \), \( 5x \), and \( 11x \), where \( x \) is a common multiplier. Since the sum of the angles in any triangle is always \( 180^\circ \), we can set up the following equation: \[ 14x + 5x + 11x = 180^\circ \] Combining the terms on the left side gives us: \[ 30x = 180^\circ \] Now, we solve for \( x \): \[ x = \frac{180^\circ}{30} = 6^\circ \] Now we can find the measures of each angle by multiplying \( x \) with each part of the ratio: 1. The first angle is \( 14x = 14 \times 6^\circ = 84^\circ \). 2. The second angle is \( 5x = 5 \times 6^\circ = 30^\circ \). 3. The third angle is \( 11x = 11 \times 6^\circ = 66^\circ \). Therefore, the measures of the angles in the triangle are \( 84^\circ \), \( 30^\circ \), and \( 66^\circ \). To double-check: - The sum is \( 84^\circ + 30^\circ + 66^\circ = 180^\circ \), confirming that our calculations are correct!