Question

find the value of ( x ) in the convex polygon.
the convex polygon (an octagon) has the following angle measures: ( 140^circ ), ( (12x + 16)^circ ), ( 137^circ ), ( (13x + 12)^circ ), ( (13x + 3)^circ ), ( (11x + 19)^circ ), ( (12x + 7)^circ ), ( (11x + 26)^circ ).

Explanation:

Step1: Determine the number of sides (n)

The polygon is an octagon, so \( n = 8 \).

Step2: Recall the formula for the sum of interior angles

The sum of interior angles of a convex polygon is \( (n - 2) \times 180^\circ \). For \( n = 8 \), the sum is \( (8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ \).

Step3: Sum all the given angles

The angles are \( 140^\circ \), \( (11x + 26)^\circ \), \( (12x + 7)^\circ \), \( (11x + 19)^\circ \), \( (13x + 3)^\circ \), \( (13x + 12)^\circ \), \( 137^\circ \), and \( (12x + 16)^\circ \).

Summing them up:

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Step4: Set the sum equal to 1080 and solve for x

$$ 360 + 72x = 1080 $$

Subtract 360 from both sides:

$$ 72x = 1080 - 360\\ 72x = 720 $$

Divide both sides by 72:

$$ x = \frac{720}{72}\\ x = 10 $$

Answer:

\( x = 10 \)