Question

the diagram shows a convex polygon.
what is the value of v?
v = \boxed{}

Explanation:

Step1: Determine the number of sides

The polygon has 6 sides, so it's a hexagon. The sum of interior angles of a hexagon is \((6 - 2)\times180^{\circ}= 720^{\circ}\).

Step2: Sum all given angles

Add the known angles and the expressions with \(v\):
\(107 + 148 + 113 + (2v - 40) + (v + 28) + (2v + 4)\).

Simplify the constants: \(107 + 148 + 113 - 40 + 28 + 4 = 107+148=255; 255 + 113 = 368; 368 - 40 = 328; 328 + 28 = 356; 356 + 4 = 360\).

Simplify the \(v\)-terms: \(2v + v + 2v = 5v\).

So the total sum is \(5v + 360\).

Step3: Set up the equation

Since the total sum of interior angles is \(720^{\circ}\), we have:
\(5v + 360 = 720\).

Subtract 360 from both sides: \(5v = 720 - 360 = 360\).

Divide by 5: \(v=\frac{360}{5}=72\).

Answer:

\(v = 72\)