Question

find the value of x. \
\
\\( \bigcirc \\ x = 2 \\)\
\\( \bigcirc \\ x = 3 \\)\
\\( \bigcirc \\ x = 33 \\)\
\\( \bigcirc \\ x = 52 \\)\
\
(diagram: collinear points q, r, s with arrows (line). triangle trs: angle at t is \\( (25x)^\circ \\), angle at s is \\( (57 + x)^\circ \\), exterior angle at r (adjacent to q-r) is \\( (45x)^\circ \\))

Explanation:

Step1: Use exterior angle theorem

The exterior angle at \( R \) (\( 45x^\circ \)) is equal to the sum of the two non - adjacent interior angles of \( \triangle TRS \), which are \( (25x)^\circ \) and \( (57 + x)^\circ \). So we have the equation:
\( 45x=25x + 57+x \)

Step2: Simplify the right - hand side

Combine like terms on the right - hand side: \( 25x+x=26x \), so the equation becomes \( 45x=26x + 57 \)

Step3: Solve for \( x \)

Subtract \( 26x \) from both sides of the equation: \( 45x-26x=26x + 57-26x \)
\( 19x=57 \)
Divide both sides by 19: \( x=\frac{57}{19}=3 \)

Answer:

\( x = 3 \) (corresponding to the option \( x = 3 \))