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## Official Guide Explanation:

Problem Solving #D10

**Background**

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Review (12th ed.). Click the links on the question number, difficulty level, and categories to find explanations for other problems.

These are the same explanations that are featured in my "Guides to the Official Guide" PDF booklets. However, because of the limitations of HTML and cross-browser compatibility, some mathematical concepts, such as fractions and roots, do not display as clearly online.

Click here for an example of the PDF booklets. Click here to purchase a PDF copy.

**Solution and Metadata**

**Question****: D10**

Page: 21

Difficulty: **7** (Very Difficult)

Category 1: Geometry > Polygons (Convex) >

Category 2: Geometry > Triangles > other

**Explanation:** The easiest way to do this is to assume that the interior pentagon consists of 5 equal angles. (It may not, but since the question is structured to imply that the sum of the 5 variables is always the same, it doesn't matter what angle measures we assign to the interior angles of the pentagon, so long as their sum is correct.)

The interior angles of a pentagon should add up to (5 - 2)(180) = 540, so if the angles are equal, each angle is 108 degrees.

Next, pick any one of the five variables. It is part of a quadrilateral containing three of the interior angles of the pentagon. Since the sum of the angles in a quadrilateral is 360, we can say:

v + 108 + 108 + 108 = 360

v + 324 = 360

v = 36

The same applies to the other four variables, so the sum of the five variables is:

5(36) = 180, choice (C).

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