Question

the convex polygon below has 7 sides. find the value of x. 92° 152° 144° 130° 142° 128° x° 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 convex 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 interior angles

We add up the known angles: \( 92^{\circ}+152^{\circ}+144^{\circ}+142^{\circ}+128^{\circ}+130^{\circ}+x^{\circ} \)
First, calculate the sum of the known angles:
\( 92 + 152+144 + 142+128+130\)
\(=(92 + 152)+(144 + 142)+(128 + 130)\)
\(=244+286 + 258\)
\(=244+(286 + 258)\)
\(=244 + 544\)
\(=788\)

Step3: Solve for \( x \)

Since the sum of all interior angles is \( 900^{\circ} \), we have the equation:
\( 788+x=900 \)
Subtract 788 from both sides:
\( x=900 - 788=112 \)

Answer:

\( x = 112 \)