Question

A nonagon has interior angles of 147 degrees, 178 degrees, 146 degrees and 193 degrees. Calculate the magnitude of each unknown interior angle given that they are equal.

Answer 1

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:

1. **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, \( n = 9 \):
\[
S = (9 - 2) \times 180^\circ = 7 \times 180^\circ = 1260^\circ
\]

2. **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:
\[
147^\circ + 178^\circ + 146^\circ + 193^\circ = 664^\circ
\]

3. **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:
\[
664^\circ + 5x = 1260^\circ
\]

4. **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:
\[
\boxed{119.2^\circ}
\]