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Question:
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°
Approach Solution 1
A regular polygon's exterior angle is equal to 360 degrees divided by the number of sides in the polygon, or n.
Exterior angle minus interior angle equals 180°.
Every side of a regular polygon is equal.
The dimensions of an ordinary polygon are the same for both interior and outer angles.
This is a typical pentagon. So, n = 5.
A regular pentagon's outside angle is equal to 360/5, or 72°.
A regular pentagon's interior angle is 180° minus 72°, or 108°.
The angle between two diagonals where they intersect at a vertex must be determined. You'll see that we are requested to determine the size of a "portion" of the inner angle if you create a pentagon and join all the diagonals.
The triangle created by two pentagon sides and a diagonal is an isosceles triangle, and the interior angle will have the biggest angle. In other words, the measure of two smaller equal angles is equal to 36°, or (180° - 108°) / 2.
The same will hold true for all other isosceles triangles (two pentagon sides and a diagonal) constructed within the conventional pentagon.
We require two of these isosceles triangles to answer this query. Each will provide a 36° smaller angle.
108° - 36° - 36° = 36° is the angle between the diagonals.
B is the correct answer.
Approach Solution 2
Sum = 180 x (n - 2) can be used to compute the interior angle measurements of a polygon with n sides. Thus, the total internal angle for a pentagon with 5 sides is 180 x (5 - 2) = 540. As a result, a regular pentagon has 108 degrees each angle, or 540/5.
Drawing a diagonal results in an isosceles triangle with three angles of 36, 36, and 108 degrees. The pentagon's angle is split into three angles, two of which are 36 degrees, when two diagonals are drawn from the same vertex. Let x be the length of the angle formed by two diagonals, and we get:
36 + x + 36 = 108
x + 72 = 108
x = 36
B is the correct answer.
Approach Solution 3
This is a GMAT Problem solving question. Here, the data given in the questions has to be analyzed to answer the question. Several areas of mathematics can be involved in the process. The way options are given is very close to correct answer, and normal guessing can lead to mistakes. Students need to understand the question properly and use proper methods to approach the answer.
The exterior angle of a regular polygon is equal to 360°/n, where n is the number of sides of the regular polygon.
180° interior angle minus 180° external angle
All sides of a regular polygon are equal.
The outer and interior angles of a regular polygon have the same measurement.
We have a standard pentagon here. So, n = 5.
A regular pentagon's outside angle is 360°/ 5 = 72°.
180° - 72° = 108° inside angle of a regular pentagon
We are asked to calculate the angle formed by two diagonals that intersect at a vertex. You'll see that we're requested to find the measure of a "portion" of the interior angle if you create a pentagon and connect all the diagonals.
The triangle formed by two sides of a pentagon and a diagonal is an isosceles triangle, with the biggest angle being the interior angle. This suggests that the sum of two smaller equal angles is 36° (180° - 108°).
The same holds true for all isosceles
Angle formed by intersecting diagonals = 108° - 36° - 36° = 36°
The appropriate answer is B.
“Regular pentagon P has all five diagonals drawn. What is the angle between two of these diagonals" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.
To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.
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