Answer:
The interior angles of the polygon are each 140°, and the exterior angles of the polygon are each 40°.
Step-by-step explanation:
We can create a system of equations to solve for both the exterior and angle measures of a regular polygon.
The first equation we can create based on the fact that the exterior and corresponding interior angle of any shape are supplementary, so their measures add to 180°:
→ [tex]I + E = 180\°[/tex]
where:
Also, remember that all of the interior and exterior angles are congruent because we are dealing with a regular polygon.
The second equation we can create based on the given information:
"each of the exterior angles ... is 100 degrees less than the [corresponding] interior angle"
→ [tex]E = I - 100\°[/tex]
The resulting system of equations is:
[tex]\begin{cases} I + E = 180\° \\ E = I - 100\° \end{cases}[/tex]
Now, we can solve for [tex]I[/tex] and [tex]E[/tex] using substitution. The second equation gives us a definition for [tex]E[/tex] in terms of [tex]I[/tex] which we can plug into the first equation:
[tex]I + E = 180\°[/tex]
↓ plugging in the second equation
[tex]I + (I - 100\°) = 180\°[/tex]
↓ combining like terms
[tex]2I - 100\° = 180\°[/tex]
↓ adding 100° to both sides
[tex]2I = 280\°[/tex]
↓ dividing both sides by 2
[tex]\boxed{I = 140\°}[/tex]
We can now plug this [tex]I[/tex]-value back into the definition for [tex]E[/tex]:
[tex]E = I - 100\°[/tex]
[tex]E = 140\° - 100\°[/tex]
[tex]\boxed{E = 40\°}[/tex]
So, the interior angles of the polygon are each 140°, and the exterior angles of the polygon are each 40°.
