Question

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

Explanation:

Step1: Find angle A in △ABC

The sum of angles in a triangle is \(180^\circ\). In △ABC, we know \(\angle B = 48^\circ\) and \(\angle C = 108^\circ\). So, \(\angle A=180^\circ - 48^\circ - 108^\circ = 24^\circ\).

Step2: Use congruent triangles property

Since \(\triangle ABC\cong\triangle FDE\), corresponding angles are equal. So, \(\angle F=\angle A = 24^\circ\). And we are given \(\angle F=(2x - y)^\circ\), but we also know from the triangle angle sum or corresponding angles (assuming y is related to another angle, but since we need to find x, and maybe y is equal to \(\angle B = 48^\circ\) as corresponding angles? Wait, maybe I misread. Wait, actually, in congruent triangles, \(\angle F=\angle A\), and also \(\angle D=\angle B = 48^\circ\), \(\angle E=\angle C = 108^\circ\). But the angle at F is \(2x - y\), and angle A is 24°, angle B is 48° (so angle D = y = 48°). Then, \(\angle F = 2x - y=24^\circ\). Substitute y = 48°? Wait, no, maybe angle F corresponds to angle A, angle D corresponds to angle B, angle E corresponds to angle C. So angle D = y = angle B = 48°, angle F = 2x - y = angle A = 24°. So substitute y = 48 into \(2x - y = 24\): \(2x - 48 = 24\).

Step3: Solve for x

Add 48 to both sides: \(2x=24 + 48=72\). Then divide by 2: \(x = 36\). Wait, no, wait, maybe I made a mistake. Wait, angle A is 24°, angle F is equal to angle A, so \(2x - y=24\). And angle D is equal to angle B, so y = 48. Then \(2x - 48 = 24\), so \(2x=72\), \(x = 36\)? Wait, but let's check again. Wait, maybe the angle at F is equal to angle A, which is 24°, and angle D is equal to angle B (48°), so y = 48. Then \(2x - y=24\), so \(2x=24 + 48=72\), so \(x = 36\). Wait, but maybe the problem is simpler. Wait, maybe the angle at F is angle A, which is 24°, and the expression for angle F is \(2x - y\), but if we assume that y is equal to angle B (48°), then solving \(2x - 48 = 24\) gives x = 36. But maybe there's a simpler way. Wait, maybe the triangle FDE has angle D = y = 48° (corresponding to angle B), angle E = 108° (corresponding to angle C), so angle F = 180 - 48 - 108 = 24°, which is equal to \(2x - y\). If y = 48, then \(2x - 48 = 24\), so \(2x = 72\), \(x = 36\).

Answer:

\(x = 36\)