Question

find ( mangle4 ).

Explanation:

Step1: Find \(m\angle3\)

Use the exterior - angle theorem (\(m\angle3+25^{\circ}=51^{\circ}\)).
\(m\angle3 = 51^{\circ}-25^{\circ}=26^{\circ}\)

Step2: Find \(m\angle1\)

Use the fact that the sum of angles in a right - triangle is \(90^{\circ}\) (\(m\angle1 + 35^{\circ}+28^{\circ}=90^{\circ}\)).
\(m\angle1=90^{\circ}-(35^{\circ} + 28^{\circ})=27^{\circ}\)

Step3: Find \(m\angle2\)

Since the sum of angles around a point on a straight line (for the angles \(\angle3,\angle2,\angle1\)) is \(180^{\circ}-90^{\circ} = 90^{\circ}\) (\(m\angle3+m\angle2+m\angle1 = 90^{\circ}\)).
\(m\angle2=90^{\circ}-(m\angle3 + m\angle1)=90^{\circ}-(26^{\circ}+27^{\circ}) = 37^{\circ}\)

Step4: Consider the right - triangle with \(\angle4\)

Use the property that the sum of angles in a triangle is \(180^{\circ}\). In the right - triangle (one angle is \(90^{\circ}\)), and we know the other non - right angle related to the previous angles.
\(m\angle4=37^{\circ}\)

Answer:

\(37^{\circ}\)