Question

in the diagram, ( \triangle abc cong \triangle fde ).
find the value of ( x ).

Explanation:

Step1: Find the value of \(y\)

Since \(\triangle ABC\cong\triangle FDE\), then \(\angle D=y^{\circ}=\angle B = 48^{\circ}\) (corresponding angles of congruent triangles are equal).

Step2: Use the angle relationship

In \(\triangle FDE\), we know that \(\angle F=(2x - y)^{\circ}\), and by the angle - sum property of a triangle (\(\angle D+\angle E+\angle F = 180^{\circ}\)). Also, \(\angle E=\angle C = 108^{\circ}\) (corresponding angles of congruent triangles).
Substitute \(y = 48\) into \(\angle F=(2x - y)^{\circ}\), and use \(\angle D+\angle E+\angle F=180^{\circ}\).
We get \(y+(2x - y)+108 = 180\).
Simplify the equation: \(48+(2x - 48)+108 = 180\), which is equivalent to \(2x+108=180\).
Subtract 108 from both sides: \(2x=180 - 108\).
So \(2x = 72\), then \(x = 36\).

Wait, there is a mistake above. Let's start over.

Since \(\triangle ABC\cong\triangle FDE\), \(\angle A=\angle F\), \(\angle B=\angle D = y = 48^{\circ}\), \(\angle C=\angle E = 108^{\circ}\).
In \(\triangle ABC\), \(\angle A=180^{\circ}-\angle B-\angle C=180-(108 + 48)=24^{\circ}\).
Since \(\angle A=\angle F\), then \(2x-y=24\).
Substitute \(y = 48\) into \(2x - y=24\).
We have \(2x-48 = 24\).
Add 48 to both sides: \(2x=24 + 48\).
\(2x=72\), so \(x = 78\).

Answer:

\(x = 78\)