Question - Solving for x in a Hexagon with Known Angles
Solution:
To solve for $$ x $$ in the hexagon shown in the image, we will first determine the sum of the interior angles of a hexagon. The sum of the interior angles of any polygon can be found using the formula:\[ S = (n - 2) \times 180^\circ \]where $$ S $$ is the sum of interior angles and $$ n $$ is the number of sides. For a hexagon ($$ n = 6 $$), this formula gives:\[ S = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ \]The problem states that three of the angles we know (120°, x, and 135°), and the remaining three interior angles are equal. Let's call each of the three unknown equal angles $$ y $$. We can set up an equation because we know that the sum of all angles must equal 720°:\[ 120^\circ + x + 135^\circ + 3y = 720^\circ \]Combining the known angles gives us:\[ 255^\circ + x + 3y = 720^\circ \]Now subtract 255° from both sides of the equation to solve for $$ x + 3y $$:\[ x + 3y = 720^\circ - 255^\circ \]\[ x + 3y = 465^\circ \]Since we do not have the individual values for $$ x $$ and $$ y $$, let's find $$ x $$ in terms of $$ y $$. Now we express $$ x $$ as:\[ x = 465^\circ - 3y \]To find the value of one of the equal angles $$ y $$, we need additional information which the question seems to omit. However, in typical hexagon problems, if the angles are not provided, it may be assumed that the hexagon is a regular hexagon, where all angles would be equal. In this case, the problem states that three angles are equal and the others are not, making it impossible to calculate $$ x $$ without further information about $$ y $$. Please check the problem again to see if there’s any missing information that would allow us to solve for $$ x $$.
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