Question

find ( mangle4 ).

Explanation:

Step1: Find \(m\angle3\)

In a triangle, the exterior angle is equal to the sum of the non - adjacent interior angles. But here, for the small triangle with angle \(25^{\circ}\) and the angle adjacent to \(51^{\circ}\) (assuming some basic triangle angle sum properties). Wait, more accurately, using the property that in a triangle, the sum of angles is \(180^{\circ}\). For the triangle with angle \(25^{\circ}\) and the other angle (let's assume we can use the fact that \(\angle3 + 25^{\circ}=51^{\circ}\) (exterior angle property of a triangle: the exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles). So \(m\angle3=51^{\circ}- 25^{\circ}=26^{\circ}\).

Step2: Use the property of right - angled triangles and parallel lines (if applicable, but mainly triangle angle sum)

We know that in a right - angled triangle (the one with \(\angle4\) and \(35^{\circ}\) and the right - angle). The sum of angles in a triangle is \(180^{\circ}\). Since we can observe that the triangle containing \(\angle4\) is a right - angled triangle (one angle is \(90^{\circ}\)). Using the angle sum property of a triangle \(A + B+C = 180^{\circ}\), where \(A = 90^{\circ}\), \(B = 35^{\circ}\), and \(C=\angle4\). So \(m\angle4=180^{\circ}-90^{\circ}-35^{\circ}=55^{\circ}\)

Answer:

\(m\angle4 = 55^{\circ}\)