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.
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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.