Question

find the value of y.

Explanation:

Step1: Use the triangle angle - sum property

In \(\triangle ABC\), \(\angle A = 65^{\circ}\), \(\angle C=90^{\circ}\). By the angle - sum property of a triangle (\(\angle A+\angle B+\angle C = 180^{\circ}\)), we can find \(\angle B\). But we can also use the fact that if two right - angled triangles are similar (by AA similarity, since \(\angle C=\angle F = 90^{\circ}\) and assume the triangles are similar in the context of the problem's visual layout).
In a right - angled triangle, the non - right angles are complementary. In \(\triangle ABC\), \(\angle A + \angle B=90^{\circ}\) (since \(\angle C = 90^{\circ}\)). In \(\triangle DEF\), \(\angle E+\angle D = 90^{\circ}\). If we assume the triangles have the same set of non - right angles (by the problem's geometric relationship, likely congruent or similar non - explicitly named right - angled triangles in a basic geometry problem setup).
We know that in a right - angled triangle, the two non - right angles add up to \(90^{\circ}\). For the left - hand triangle \(\triangle ABC\) with \(\angle A = 65^{\circ}\) and \(\angle C=90^{\circ}\), and for the right - hand triangle \(\triangle DEF\) with \(\angle F = 90^{\circ}\).
Since the non - right angles in right - angled triangles (when there is an implicit geometric relationship like congruence or a basic angle - matching problem in geometry) and we can directly equate the angles.

Step1: Recall the angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). For a right - angled triangle (\(\angle = 90^{\circ}\)), the sum of the two non - right angles is \(180^{\circ}-90^{\circ}=90^{\circ}\).

Step2: Calculate \(y\)

In \(\triangle ABC\), \(\angle A = 65^{\circ}\), \(\angle C = 90^{\circ}\). In \(\triangle DEF\), \(\angle F=90^{\circ}\). Using the fact that for right - angled triangles \(\angle A+\angle B = 90^{\circ}\) and \(\angle E+\angle D=90^{\circ}\). If we assume the non - named congruent or similar relationships (basic angle - finding in right - angled triangles), \(y=90^{\circ}-65^{\circ}\)

Answer:

\(y = 25\)

Correct process: