Question

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

Explanation:

Step1: Find total interior angles

First, count the sides: this is a 7-sided polygon. The formula for total interior angles of an n-sided polygon is $(n-2)\times180^\circ$.
$$(7-2)\times180^\circ = 5\times180^\circ = 900^\circ$$

Step2: Sum all given angles

Add up all the angle expressions and known angles:
$$2c + (2c+34^\circ) + 2c + 2c + 148^\circ + 146^\circ + (c+50^\circ)$$
Combine like terms:
$$2c+2c+2c+2c+c + 34^\circ+148^\circ+146^\circ+50^\circ = 9c + 378^\circ$$

Step3: Set up and solve equation

Set the total equal to 900° and solve for c:
$$9c + 378^\circ = 900^\circ$$
Subtract $378^\circ$ from both sides:
$$9c = 900^\circ - 378^\circ = 522^\circ$$
Divide by 9:
$$c = \frac{522^\circ}{9} = 58^\circ$$

Answer:

$58$