That figure has 45 sides.
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The easiest way to solve this is to use the exterior angles of the polygon. The sum of an exterior angle and interior angle is 180o. Also the sum of all the exterior angles of a polygon is 360o. As the polygon is regular, all the interior angles are the same and all the exterior angles are the same. So, for an interior angle of 172o the exterior angle is 180-172 = 8o. The sum of all the exterior angles is 360o so how many 8o make up 360o? 360/8=45. Thus the polygon with interior angles of 172o has 45 sides.
The name of a 172 sided polygon is ChiliHeptacontakaidigon.
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Supplement of an angle measure x* be y*=(180-x*). Or x+y=180* for the problem x=9y => x+y = 9y+y=10y=180 => y=18* and x =172*