Question

select the correct answer.
in the diagram, transversal ( t ) cuts parallel lines ( a ) and ( b ). which equation is necessarily true?
( \bigcirc ) a. ( m angle 1 = m angle 7 )
( \bigcirc ) b. ( m angle 3 = m angle 6 )
( \bigcirc ) c. ( m angle 5 + m angle 8 = 90 ^ { circ } )
( \bigcirc ) d. ( m angle 6 + m angle 7 = 180 ^ { circ } )

Explanation:

Step1: Recall the properties of parallel lines cut by a transversal

When a transversal cuts parallel lines, alternate - interior angles, corresponding angles, and consecutive - interior angles have specific relationships.

Step2: Analyze each option

• Option A:

\(m\angle1\) and \(m\angle7\): \(\angle1\) and \(\angle5\) are corresponding angles (\(m\angle1 = m\angle5\) since \(a\parallel b\)), and \(\angle5\) and \(\angle7\) are vertical angles (\(m\angle5=m\angle7\)). By the transitive property, \(m\angle1 = m\angle7\).

• Option B:

\(m\angle3\) and \(m\angle6\): \(\angle3\) and \(\angle5\) are consecutive - interior angles. \(m\angle3 + m\angle5=180^{\circ}\) (since \(a\parallel b\)), and \(m\angle5 = m\angle6\) (vertical angles). So \(m\angle3+m\angle6 = 180^{\circ}\), not \(m\angle3=m\angle6\).

• Option C:

\(m\angle5\) and \(m\angle8\): \(\angle5\) and \(\angle8\) are vertical angles. \(m\angle5=m\angle8\), and there is no reason for \(m\angle5 + m\angle8=90^{\circ}\) (they are equal, but their sum is \(2m\angle5\) which is not necessarily \(90^{\circ}\)).

• Option D:

\(m\angle6\) and \(m\angle7\): \(\angle6\) and \(\angle7\) are vertical angles. \(m\angle6=m\angle7\), and \(m\angle6 + m\angle7 = 2m\angle6
eq180^{\circ}\) (they are equal, not supplementary).

Answer:

A. \(m\angle1 = m\angle7\)