Official Guide Explanation:
Problem Solving #D10

Background

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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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