23. The angles of a triangle are in ratio \( 2: 3 \) and the third is \( \frac{4 \pi^{c}}{9} \). Find the all angles in degrees.

To solve for the angles of the triangle, let's denote the angles as \( 2x \), \( 3x \), and the third angle \( \frac{4\pi^c}{9} \). We know that the sum of the angles in a triangle is \( 180^\circ \). First, we convert \( \frac{4\pi^c}{9} \) from radians to degrees. Since \( \pi \) radians is equal to \( 180^\circ \), we can find the equivalent degree measure: \[ \frac{4\pi^c}{9} \text{ radians} = \frac{4\pi^c \times 180^\circ}{\pi} = \frac{720^c}{9} = 80^c. \] Now, we can set up the equation for the sum of angles: \[ 2x + 3x + 80^c = 180^\circ. \] Combining the terms gives: \[ 5x + 80^c = 180^\circ. \] From this, we can isolate \( 5x \): \[ 5x = 180^\circ - 80^c. \] Next, we can find the measure of \( x \): \[ x = \frac{180^\circ - 80^c}{5}. \] Now we can substitute \( x \) back into the expressions for the angles: 1. First angle: \( 2x = 2 \left( \frac{180^\circ - 80^c}{5} \right) = \frac{360^\circ - 160^c}{5} \). 2. Second angle: \( 3x = 3 \left( \frac{180^\circ - 80^c}{5} \right) = \frac{540^\circ - 240^c}{5} \). 3. The third angle: \( 80^c \). Thus, the angles of the triangle are: - Angle 1: \( \frac{360 - 160^c}{5} \) degrees. - Angle 2: \( \frac{540 - 240^c}{5} \) degrees. - Angle 3: \( 80^c \) degrees. This gives a complete set of angles for the triangle based on the provided ratios and third angle.