Question

the convex polygon below has 7 sides. find the value of x. the polygon has angles: 106°, 139°, 126°, x°, 141°, 112°, 129°

Explanation:

Step1: Recall polygon angle sum formula

For a convex polygon with \( n \) sides, the sum of interior angles is \( (n - 2)\times180^\circ \). Here, \( n = 7 \), so sum is \( (7 - 2)\times180^\circ = 5\times180^\circ = 900^\circ \).

Step2: Sum the known angles

Known angles: \( 106^\circ, 139^\circ, 126^\circ, x^\circ, 141^\circ, 112^\circ, 129^\circ \). Sum of known (excluding \( x \)): \( 106 + 139 + 126 + 141 + 112 + 129 = 753 \).

Step3: Solve for \( x \)

Let sum of all angles be \( 900^\circ \). So \( 753 + x = 900 \). Then \( x = 900 - 753 = 147 \).

Answer:

\( x = 147 \)