Question

the convex polygon below has 7 sides. find the value of x.
139° x°
106° 147°
129° 141°
112°
x =

Explanation:

Step1: Recall the formula for the sum of interior angles of a polygon.

The formula for the sum of the interior angles of a polygon with \( n \) sides is \( S=(n - 2)\times180^{\circ} \). For a 7 - sided polygon (\( n = 7 \)), we calculate the sum:
\( S=(7 - 2)\times180^{\circ}=5\times180^{\circ} = 900^{\circ} \)

Step2: Sum the given angles.

We add up the known angles: \( 106^{\circ}+139^{\circ}+129^{\circ}+112^{\circ}+141^{\circ}+147^{\circ} \)
First, \( 106 + 139=245 \); \( 245+129 = 374 \); \( 374+112=486 \); \( 486+141 = 627 \); \( 627+147=774 \)

Step3: Solve for \( x \).

Let the sum of all interior angles be \( 900^{\circ} \). Let the sum of known angles be \( 774^{\circ} \). Then \( x+774 = 900 \)
Subtract \( 774 \) from both sides: \( x=900 - 774=126 \)

Answer:

\( x = 126 \)