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06 Sep 2016, 17:12
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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°
(A) 12°
(B) 36°
(C) 54°
(D) 60°
(E) 72°
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.
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.
