Question

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

Explanation:

Step1: Recall exterior angle sum rule

For any convex polygon, the sum of all exterior angles is $360^\circ$.

Step2: Sum all given exterior angles

$$(x-1) + 35 + 2x + (3x-46) + (2x-31) + (2x-31) + 4x = 360$$

Step3: Simplify left-hand side

Combine like terms:
$$x-1+35+2x+3x-46+2x-31+2x-31+4x = (x+2x+3x+2x+2x+4x) + (-1+35-46-31-31)$$
$$= 14x - 74$$

Step4: Solve for x

Set equal to 360 and solve:
$$14x - 74 = 360$$
$$14x = 360 + 74$$
$$14x = 434$$
$$x = \frac{434}{14} = 31$$

Answer:

$31$