regular polygon has an interior angle measure of 150°. How many diagonals does this polygon have? Please explain.
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The sum of the exterior angles of a regular polygon is 360, one angle per vertex. For each exterior angle, there is an adjacent interior angle of the regular polygon. The two angles sum to 180 because they are supplementary.
The exterior angle in this problem has measure 30 [180 - 150 = 30].
The sum of the exterior angles is 360. 360/30 = 12. So, the regular polygon has 12 sides and is a dodecagon.
The number of diagonals of a 12-gon [another name for dodecagon] can be found by drawing them and then counting. There is a formula that can be derived, a formula that gives the number of diagonals of a convex n-gon.
It is the following: [n(n - 3)] / 2. For the dodecagon, it would give [12(9)] / 2 = 54.
Thank you so much!
Glad to help. Thanks for the medal.
I answered your question about lines m and n also.