Topic: Ratios the angles of triangles are in a ratios \( 1: 3: 8 \). What are the angles.

To determine the measures of the angles in a triangle where the angles are in the ratio \(1 : 3 : 8\), follow these steps: 1. **Understand the Ratio:** Let the angles be represented as: \[ \text{Angle}_1 = x \] \[ \text{Angle}_2 = 3x \] \[ \text{Angle}_3 = 8x \] 2. **Use the Triangle Angle Sum Property:** The sum of the interior angles of any triangle is always \(180^\circ\). Therefore: \[ x + 3x + 8x = 180^\circ \] 3. **Combine Like Terms:** \[ 12x = 180^\circ \] 4. **Solve for \(x\):** \[ x = \frac{180^\circ}{12} = 15^\circ \] 5. **Find Each Angle:** Now that \(x = 15^\circ\), substitute back to find each angle: \[ \text{Angle}_1 = x = 15^\circ \] \[ \text{Angle}_2 = 3x = 3 \times 15^\circ = 45^\circ \] \[ \text{Angle}_3 = 8x = 8 \times 15^\circ = 120^\circ \] 6. **Conclusion:** The measures of the angles in the triangle are: \[ 15^\circ,\ 45^\circ,\ \text{and}\ 120^\circ \]