Reasoning · Grade 4-2 Polygons and Angles
Problem
Draw the erased inside lines back in.
Nothing about the star has changed - the lines were only rubbed out. Putting them back turns a strange 20-sided outline into ten copies of a shape whose angle rules you already know.
4.G.A.1Draw A DiagramA smaller star shows the pattern first.
Testing the idea on a case you can picture in your head - four squares round a point - proves the method before you trust it on ten.
4.MD.C.7Solve An Easier Related ProblemThe centre angle is 360 ÷ 10 = 36 degrees.
Angles around a point add up like pieces of a pie; ten equal slices of a full turn is a division you can do in one step.
4.MD.C.7Identify SubproblemsThe ten rhombuses share the centre, so each one's centre angle is one tenth of a full turn.
The ten centre angles fill one complete turn around the centre with no gap and no overlap.
Because the ten rhombuses are identical, the turn is cut into ten equal shares and each angle is one of them.
The tip is its opposite, so 36 degrees.
You can see it without any formula: turning the rhombus half a turn about its middle lands it exactly on itself, and that swap carries the centre corner onto the tip corner, so those two angles must match.
8.G.A.5Identify SubproblemsThe rhombus's large angle is 144 degrees.
Slide along one side of the rhombus from one corner to the next: the two sides you leave and arrive on are parallel, so the two angles together straighten out into a half turn.
7.G.B.5Identify SubproblemsAt the notch the second angle is 72 degrees.
A full turn is 360 degrees no matter where you stand. Once you know how much of the turn is covered by the shapes, the leftover gap is a single subtraction.
4.MD.C.7Identify SubproblemsDraw the rubbed-out lines back in: ten equal shapes sharing one full turn means 360 divided by 10, and every other angle in the star follows from that one number.