Question

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the diagram shows a convex polygon.
what is the value of c?
$c = \square^\circ$

Explanation:

Step1: Recall the exterior angle sum of a polygon

The sum of the exterior angles of any convex polygon is \(360^\circ\). So we can set up an equation by adding all the given exterior angles and setting it equal to \(360^\circ\).
The exterior angles are \(c\), \(52^\circ\), \(2c + 4^\circ\), \(c - 2^\circ\), and \(3c - 23^\circ\). So the equation is:
\(c + 52+(2c + 4)+(c - 2)+(3c - 23)=360\)

Step2: Simplify the left - hand side of the equation

First, combine like terms:
\(c+2c + c+3c+52 + 4- 2-23=360\)
\((1 + 2+1 + 3)c+(52 + 4-2 - 23)=360\)
\(7c+(56 - 25)=360\)
\(7c + 31=360\)

Step3: Solve for \(c\)

Subtract 31 from both sides of the equation:
\(7c=360 - 31\)
\(7c=329\)
Then divide both sides by 7:
\(c=\frac{329}{7}\)
\(c = 47\)

Answer:

\(47\)