Sum of largest two angles

Question

All the inner angles of a 7 sided polygon are obtuse, their sizes in degrees being distinct integers divisible by 9. What is the sum (in degree) of the largest two angles?

Answer 1

Sum of all angles is $180^\circ \cdot (n-2) = 900^\circ$.

Denote angles as $a,b,c,d,e,f,g$ ($a<b<c<d<e<f<g$).

If all angles are obtuse and are integer numbers divisible by $9^\circ$, then they can be $99^\circ, 108^\circ, 117^\circ, 126^\circ, 135^\circ, 144^\circ, 153^\circ, ...$

A). If $a\ge 108^\circ$, then $b\ge 117^\circ, c\ge 126^\circ, d\ge 135^\circ, \ldots$, then $a+b+c+d+e+f+g\ge (108+117+126+135+144+153+162)^\circ = 945^\circ>900^\circ.$

So, $a=99^\circ$.

B). If $b\ge 117^\circ$, then $c\ge 126^\circ$, $\ldots$, then $a+b+c+d+e+f+g\ge (99+117+126+135+144+153+162)^\circ= 936^\circ>900^\circ$.

So, $b=108^\circ$.

C). Same way we can show, that $c=117^\circ$, ...

After that you can find possible values for $2$ largest angles.