Question

two parallel lines are cut by a transversal as shown below. suppose ( mangle6 = 139^{circ} ). find ( mangle1 ) and ( mangle4 ).

Explanation:

Step1: Find \(m\angle4\)

Since \(\angle4\) and \(\angle6\) are alternate interior angles. When two parallel lines are cut by a transversal, alternate interior angles are congruent. So \(m\angle4 = m\angle6\).
Given \(m\angle6 = 139^{\circ}\), then \(m\angle4=139^{\circ}\)

Step2: Find \(m\angle1\)

Since \(\angle1\) and \(\angle4\) are supplementary angles (they form a linear pair). The sum of the measures of supplementary angles is \(180^{\circ}\).
Let \(m\angle1=x\), then \(x + m\angle4=180^{\circ}\)
Substitute \(m\angle4 = 139^{\circ}\) into the equation: \(x+139^{\circ}=180^{\circ}\)
Solve for \(x\): \(x=180^{\circ}- 139^{\circ}=41^{\circ}\)

Answer:

\(m\angle1 = 41^{\circ}\) and \(m\angle4 = 139^{\circ}\)