Question

there is a triangle - like figure with some markings. one side is marked with length ( x ), another segment inside is marked 10. angles are marked as ( 4y^{circ} ), ( 7y + 5^{circ} ), ( 107^{circ} ). there are some congruent markings on sides and segments. we need to find the value of ( x ), with ( x=square ).

Explanation:

Step1: Solve for y using angle relation

The angle \(7y + 5^\circ\) and \(107^\circ\) are supplementary (they form a linear pair).
\[7y + 5 + 107 = 180\]
\[7y + 112 = 180\]
\[7y = 180 - 112\]
\[7y = 68\]
\[y = \frac{68}{7}\]

Step2: Verify triangle similarity/proportional sides

The figure shows similar triangles (marked congruent segments and parallel sides, so sides are proportional). The side \(x\) corresponds to the side of length 10, and the base segments are congruent, meaning the larger triangle has sides twice the length of the smaller one.
\[x = 2 \times 10\]

Answer:

20