Question

in the diagram, \\( \triangle a b c \cong \triangle f d e \\).
find the value of \\( x \\).

Explanation:

Step1: Find the value of \( y \)

Since \( \triangle ABC\cong\triangle FDE \), corresponding angles are equal. So \( y = 48\) (corresponding angles of congruent triangles).

Step2: Use the angle - sum property of a triangle

In \( \triangle FDE \), the sum of angles in a triangle is \( 180^{\circ}\).
We know that one angle is \( y = 48^{\circ}\), and another angle is \( (2x - y)^{\circ}\), and the third angle (corresponding to \( \angle C\)) is \( 108^{\circ}\).
So \( 108+(2x - y)+y=180\).
Substitute \( y = 48\) into the equation: \( 108+(2x - 48)+48 = 180\).
Simplify the left - hand side: \( 108+2x-48 + 48=180\), which gives \( 108+2x=180\).
Subtract \( 108\) from both sides: \( 2x=180 - 108\).
\( 2x=72\).
Divide both sides by \( 2\): \( x = 36\).

Answer:

\( x = 36\)