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Regular pentagon P has all five diagonals drawn. What is the angle bet

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Magoosh GMAT Instructor
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Regular pentagon P has all five diagonals drawn. What is the angle bet  [#permalink]

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New post 06 Sep 2016, 17:12
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A
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Difficulty:

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Question Stats:

59% (01:45) correct 41% (01:50) wrong based on 77 sessions

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Regular pentagon P has all five diagonals drawn. What is the angle between two of these diagonals where they meet at a vertex of the pentagon?

(A) 12°
(B) 36°
(C) 54°
(D) 60°
(E) 72°

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Mike McGarry
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Re: Regular pentagon P has all five diagonals drawn. What is the angle bet  [#permalink]

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New post 06 Sep 2016, 18:33
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Few important points that we need to solve this question:

Exterior angle of a regular polygon = 360°/ n, n = number of sides of the regular polygon
Interior angle = 180° - exterior angle
Regular polygon has all sides equal
Regular polygon has same measure for each of the exterior and interior angles


Here, we have a regular pentagon. So, n = 5.

Exterior angle of a regular pentagon = 360°/ 5 = 72°
Interior angle of a regular pentagon = 180° - 72° = 108°

We are asked to find the angle between two diagonals where they meet at a vertex. If you draw a pentagon and connect all the diagonals, you will notice that we are asked to find the measure of a "portion" of the interior angle.

Next, note that the triangle formed with two sides of the pentagon and a diagonal is an isosceles triangle and the largest angle will be the interior angle. This means that measure of two smaller equal angles = (180° - 108°) / 2 = 36°.

Same will apply to all such isosceles triangles (two sides of the pentagon and a diagonal) formed within the regular pentagon.

For this question, we need two such isosceles triangles. Each will contribute one smaller angle, 36°.

Angle between diagonals = 108° - 36° - 36° = 36°

Answer (B).

Due to lack of visuals, the explanation may look complicated. If you draw and follow the story, it may make sense.
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Re: Regular pentagon P has all five diagonals drawn. What is the angle bet  [#permalink]

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New post 07 Sep 2016, 19:28
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This makes it a little easier to see.
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pentagon.png [ 35.46 KiB | Viewed 1586 times ]

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Re: Regular pentagon P has all five diagonals drawn. What is the angle bet  [#permalink]

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New post 23 Jan 2018, 14:20
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Re: Regular pentagon P has all five diagonals drawn. What is the angle bet   [#permalink] 23 Jan 2018, 14:20
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