Question

1. find the measures of \\( \angle c \\) and \\( \angle f \\).

Explanation:

Step1: Prove the triangles are congruent

By the Hypotenuse - Leg (HL) congruence criterion (right - angled triangles with hypotenuse \(10\) and one leg \(8\) or \(6\) equal), \(\triangle ABC\cong\triangle EDF\).

Step2: Set up the equation for angle measures

Since \(\triangle ABC\cong\triangle EDF\), \(\angle C=\angle F\). So \(2x - 36=4x-108\).

Step3: Solve for \(x\)

Subtract \(2x\) from both sides: \(-36 = 2x-108\).
Add \(108\) to both sides: \(2x=108 - 36=72\).
Divide by \(2\): \(x = 36\).

Step4: Find \(\angle C\) and \(\angle F\)

Substitute \(x = 36\) into \(2x-36\): \(2\times36-36=72 - 36=36^{\circ}\).

Answer:

\(\angle C = 36^{\circ}\) and \(\angle F=36^{\circ}\)