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:
\[

$$\begin{align*} &140 + (11x + 26) + (12x + 7) + (11x + 19) + (13x + 3) + (13x + 12) + 137 + (12x + 16)\\ =& 140 + 26 + 7 + 19 + 3 + 12 + 137 + 16 + (11x + 12x + 11x + 13x + 13x + 12x)\\ =& (140 + 26 + 7 + 19 + 3 + 12 + 137 + 16) + (11x + 12x + 11x + 13x + 13x + 12x)\\ =& 360 + (11 + 12 + 11 + 13 + 13 + 12)x\\ =& 360 + 72x \end{align*}$$

\]

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:

\( x = 10 \)