Question

for items 2 - 5, lines ( ell ) and ( m ) are intersected by transversal ( t ). ( ellparallel m ) 2. select all the angles that are supplementary to ( angle2 ). ( square a. angle3 ) ( square c. angle5 ) ( square e. angle7 ) ( square b. angle4 ) ( square d. angle6 ) ( square f. angle8 )

Explanation:

Step1: Recall the definition of supplementary angles

Supplementary angles are two angles whose sum is \(180^{\circ}\).

Step2: Analyze adjacent angles

\(\angle2\) and \(\angle1\) form a linear - pair (adjacent angles on a straight line), so \(\angle2+\angle1 = 180^{\circ}\). Also, \(\angle2\) and \(\angle4\) form a linear - pair (\(\angle2+\angle4=180^{\circ}\)) since they are adjacent angles formed by the intersection of line \(t\) and line \(\ell\).

Step3: Use the property of parallel lines (\(\ell\parallel m\))

By the property of parallel lines and transversal (corresponding angles, alternate interior angles, etc.), \(\angle2\) and \(\angle5\) are same - side interior angles. For parallel lines \(\ell\) and \(m\) cut by transversal \(t\), \(\angle2+\angle5 = 180^{\circ}\) (same - side interior angles are supplementary).
\(\angle2\) and \(\angle7\): \(\angle2=\angle7\) (alternate exterior angles, \(\ell\parallel m\)), so they are not supplementary. \(\angle2\) and \(\angle3\): \(\angle2=\angle3\) (vertical angles), so they are not supplementary. \(\angle2\) and \(\angle6\): \(\angle2\) and \(\angle6\) are alternate interior angles (\(\angle2=\angle6\) because \(\ell\parallel m\)), so they are not supplementary. \(\angle2\) and \(\angle8\): \(\angle2\) and \(\angle8\) are corresponding angles (\(\angle2=\angle8\) because \(\ell\parallel m\)), so they are not supplementary.

Answer:

B. \(\angle4\), C. \(\angle5\)