Question

if ( mangle4 = 35^{circ} ), find ( mangle2 ) and ( mangle3 ).

Explanation:

Step1: Find \(m\angle2\)

Since \(AB\parallel CD\), \(\angle2\) and \(\angle4\) are alternate - interior angles.
By the Alternate - Interior Angles Theorem, \(m\angle2 = m\angle4\).
Given \(m\angle4 = 35^{\circ}\), so \(m\angle2=35^{\circ}\).

Step2: Find \(m\angle3\)

We know that \(\angle1 = 90^{\circ}\) (right - angle), and \(\angle1+\angle3+\angle4 = 180^{\circ}\) (straight - line angle).
Substitute \(\angle1 = 90^{\circ}\) and \(\angle4 = 35^{\circ}\) into the equation: \(90^{\circ}+m\angle3 + 35^{\circ}=180^{\circ}\).
Simplify the left - hand side: \(m\angle3+125^{\circ}=180^{\circ}\).
Subtract \(125^{\circ}\) from both sides: \(m\angle3=180^{\circ}- 125^{\circ}\).

Answer:

\(m\angle2 = 35^{\circ}\)
\(m\angle3 = 55^{\circ}\)