Question

in the diagram, \\( \triangle abc \cong \triangle fde \\).

1. find the value of \\( x \\).
2. find the value of \\( y \\).

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle ABC\cong\triangle FDE\), corresponding angles are equal. So, \(\angle B=\angle D = y^{\circ}\), \(\angle C=\angle E = 48^{\circ}\), \(\angle A=\angle F=(2x - y)^{\circ}\).
In \(\triangle ABC\), by the angle - sum property of a triangle (\(\angle A+\angle B+\angle C = 180^{\circ}\)), we know \(\angle A=180^{\circ}-\angle B-\angle C\). Substituting \(\angle B = y\) and \(\angle C = 48^{\circ}\), \(\angle A=180 - y-48=(132 - y)^{\circ}\). Also, \(\angle A=(2x - y)^{\circ}\).

Step2: Find the value of \(y\)

Since \(\angle B\) and \(\angle D\) are corresponding angles (from \(\triangle ABC\cong\triangle FDE\)), and in \(\triangle ABC\), \(\angle B = 180^{\circ}-\angle A-\angle C\). Given \(\angle A = 108^{\circ}\), \(\angle C = 48^{\circ}\), then \(y=\angle B=180-(108 + 48)=24\).

Step3: Find the value of \(x\)

Substitute \(y = 24\) into the equation \(\angle A=(2x - y)^{\circ}\). Since \(\angle A = 108^{\circ}\), we have \(108=2x-24\).
Add \(24\) to both sides of the equation: \(108 + 24=2x\), so \(132 = 2x\).
Divide both sides by \(2\): \(x=\frac{132}{2}=66\).

Answer:

1. \(x = 66\)
2. \(y = 24\)