The five exterior angles of a convex pentagon are in the ratio 1 : 2 : 3 : 4 : 5. Find the five interior angles and the ratio between them.

Step-by-step solution

1. 1

   Start from the exterior angles, not the interior ones, because their sum is the fact that does not depend on the number of sides. Walking once around any convex polygon turns you through one full revolution, so

   $\text{sum of exterior angles}=360^\circ$

   for a triangle, a pentagon or a hundred-gon alike. That single constant is what makes the given ratio usable.

2. 2

   Introduce a scale factor and solve. Write the exterior angles as $k,2k,3k,4k,5k$. Then

   $k+2k+3k+4k+5k=15k=360^\circ$

   $k=24^\circ$

3. 3

   List the exterior angles.

   $24^\circ,\quad 48^\circ,\quad 72^\circ,\quad 96^\circ,\quad 120^\circ$

   All five are less than $180^\circ$ and positive, so a convex pentagon with these angles is geometrically possible.

4. 4

   Convert each exterior angle to its interior angle. At each vertex the interior and exterior angles form a straight line, so they are supplementary:

   $\text{interior}=180^\circ-\text{exterior}$

   $180-24=156^\circ,\quad 180-48=132^\circ,\quad 180-72=108^\circ,\quad 180-96=84^\circ,\quad 180-120=60^\circ$

5. 5

   Check against the interior angle sum. For an $n$-gon the interior angles total $(n-2)\cdot180^\circ$, here $(5-2)\cdot180^\circ=540^\circ$:

   $156+132+108+84+60=540\;\checkmark$

   This is an independent confirmation, since it uses a formula that was never fed into the calculation.

6. 6

   Reduce the interior angles to a ratio. Dividing $156:132:108:84:60$ by their common factor $12$ gives

   $13:11:9:7:5.$

   Notice the ratio is not $5:4:3:2:1$ — subtracting from $180^\circ$ is not a proportional operation, so an arithmetic-progression ratio of exterior angles becomes a different arithmetic-progression ratio of interior angles. That trap is the point of the problem.