Question

transversal \overleftrightarrow{ef} cuts parallel lines \overleftrightarrow{ab} and \overleftrightarrow{cd} as shown in the diagram, and m\angle4 = 55.1^{\circ}. what are m\angle5 and m\angle7?

a. m\angle5 = 34.9^{\circ}, and m\angle7 = 145.1^{\circ}.
b. m\angle5 = 55.1^{\circ}, and m\angle7 = 34.9^{\circ}.
c. m\angle5 = 124.9^{\circ}, and m\angle7 = 55.1^{\circ}.
d. m\angle5 = 55.1^{\circ}, and m\angle7 = 124.9^{\circ}.
e. m\angle5 = 34.9^{\circ}, and m\angle7 = 55.1^{\circ}.

Explanation:

Step1: Use the property of alternate interior angles

Since \(\overleftrightarrow{AB}\parallel\overleftrightarrow{CD}\) and \(\overleftrightarrow{EF}\) is a transversal, \(\angle4\) and \(\angle5\) are alternate interior angles.
By the Alternate - Interior Angles Theorem, \(m\angle5 = m\angle4\).
So, \(m\angle5=55.1^{\circ}\).

Step2: Use the property of vertical angles

\(\angle7\) and \(\angle5\) are supplementary (they form a linear pair). Also, \(\angle7\) and \(\angle3\) are vertical angles. \(\angle3\) and \(\angle4\) are supplementary (\(\angle3+\angle4 = 180^{\circ}\)).
Since \(m\angle4 = 55.1^{\circ}\), then \(m\angle7=180^{\circ}-m\angle5\).
Substitute \(m\angle5 = 55.1^{\circ}\) into the formula: \(m\angle7=180 - 55.1=124.9^{\circ}\).

Answer:

D. \(m\angle5 = 55.1^{\circ}\), and \(m\angle7 = 124.9^{\circ}\)