Question

the diagram shows a convex polygon. what is the value of u? u = \square^\circ (angles in diagram: u, 31°, u−35°, u−41°, 2u−45°)

Explanation:

Step1: Recall the sum of exterior angles of a convex 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 equating it to \(360^\circ\). The angles are \(u\), \(31^\circ\), \(u - 35^\circ\), \(u - 41^\circ\), and \(2u - 45^\circ\). So the equation is:
\[u + 31+(u - 35)+(u - 41)+(2u - 45)=360\]

Step2: Simplify the left - hand side of the equation.

First, combine like terms:
\[u+u + u+2u+31- 35-41 - 45=360\]
\[5u+(31-(35 + 41+45))=360\]
Calculate the sum inside the parentheses: \(35 + 41+45 = 121\)
Then, \(31-121=-90\)
So the equation becomes:
\[5u-90 = 360\]

Step3: Solve for \(u\).

Add \(90\) to both sides of the equation:
\[5u-90 + 90=360 + 90\]
\[5u=450\]
Divide both sides by \(5\):
\[u=\frac{450}{5}=90\]

Answer:

\(90\)