To find the unknown angles in the nonagon where four of the interior angles are provided and the other five are equal, we can follow these steps:
-
Calculate the total sum of the interior angles of a nonagon: The formula for the sum of the interior angles \( S \) of a polygon with \( n \) sides is: \[ S = (n - 2) \times 180^\circ \] For a nonagon, \( n = 9 \): \[ S = (9 - 2) \times 180^\circ = 7 \times 180^\circ = 1260^\circ \]
-
Sum the known interior angles: We have the following interior angles:
- \( 147^\circ \)
- \( 178^\circ \)
- \( 146^\circ \)
- \( 193^\circ \)
The sum of these known angles is: \[ 147^\circ + 178^\circ + 146^\circ + 193^\circ = 664^\circ \]
-
Determine the sum of the unknown angles: Let \( x \) be the measure of each of the five equal unknown angles. The total sum of the angles can be expressed as: \[ 664^\circ + 5x = 1260^\circ \]
-
Solve for \( x \): Subtract \( 664^\circ \) from both sides: \[ 5x = 1260^\circ - 664^\circ = 596^\circ \] Now, divide by 5: \[ x = \frac{596^\circ}{5} = 119.2^\circ \]
Thus, the magnitude of each unknown interior angle is: \[ \boxed{119.2^\circ} \]