Question

the convex polygon below has 7 sides. find the value of x.

Explanation:

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

The sum of the interior angles of a convex polygon with \( n \) sides is given by the formula \( S=(n - 2)\times180^{\circ} \). 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.

The given angles are \( 130^{\circ},152^{\circ},92^{\circ},x^{\circ},142^{\circ},112^{\circ},128^{\circ} \). Let's sum the known angles:
\( 130 + 152+92 + 142+112+128\)
First, \( 130+152 = 282 \); \( 282+92=374 \); \( 374 + 142=516 \); \( 516+112 = 628 \); \( 628+128 = 756 \)

Step3: Solve for \( x \).

We know that the sum of all interior angles is \( 900^{\circ} \). Let the sum of known angles be \( 756^{\circ} \) and the unknown angle be \( x \). So:
\( 756+x=900 \)
Subtract 756 from both sides: \( x = 900 - 756=144 \)

Answer:

\( 144 \)