To determine the number of sides in a polygon when given specific angle measures, we can use the formula for the sum of interior angles of a polygon. The formula is:
Sum of interior angles = (n – 2) × 180°
where n represents the number of sides in the polygon.
In this scenario, we have:
- Four angles of 100 degrees each, contributing
4 × 100° = 400° - Let’s assume there are m angles measuring 160 degrees each. Thus, these angles contribute
160m°
The total sum of the angles can therefore be expressed as:
400° + 160m°
Setting this equal to the sum of interior angles formula, we have:
400 + 160m = (n - 2) × 180
Next, we know that the total number of angles in the polygon is the same as the number of sides, hence:
n = 4 + m
Now, substituting n in the angle sum equation:
400 + 160m = ((4 + m) - 2) × 180
Which simplifies to:
400 + 160m = (m + 2) × 180
Expanding the right-hand side gives:
400 + 160m = 180m + 360
Rearranging terms yields:
400 - 360 = 180m - 160m
Therefore:
40 = 20m
Solving for m results in:
m = 2
Now, substituting back to find n:
n = 4 + 2 = 6
Thus, the polygon has 6 sides, making it a hexagon.