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 ( angle 1 ). a. ( angle 3 ) c. ( angle 5 ) e. ( angle 7 ) b. ( angle 4 ) d. ( angle 6 ) f. ( angle 8 ) 3. select all the angles that are congruent to ( angle 5 ). a. ( angle 1 ) c. ( angle 3 ) e. ( angle 7 ) b. ( angle 2 ) d. ( angle 4 ) f. ( angle 8 ) 4. complete the following plan to prove that ( angle 3congangle 6 ). use the same - side interior angles postulate to show that ( angle 3 ) is supplementary to. show ( angle 6 ) and are because they form a. since ( angle 3 ) and ( angle 6 ) are to the same angle, they are congruent to each other.

Explanation:

Step1: Supplementary Angles

Supplementary angles add up to \(180^{\circ}\). \(\angle1\) and \(\angle2\) form a linear pair (\(\angle1+\angle2 = 180^{\circ}\)), \(\angle1\) and \(\angle4\) form a linear pair (\(\angle1+\angle4=180^{\circ}\)). Also, since \(\ell\parallel m\), \(\angle1\cong\angle5\) (corresponding angles). \(\angle5\) and \(\angle6\) form a linear pair (\(\angle5 + \angle6=180^{\circ}\)), so \(\angle1+\angle6 = 180^{\circ}\).

Step2: Congruent Angles

For congruent angles to \(\angle5\): \(\angle1\cong\angle5\) (corresponding angles as \(\ell\parallel m\)), \(\angle3\cong\angle5\) (alternate - interior angles as \(\ell\parallel m\)), \(\angle7\cong\angle5\) (vertical angles).

Step3: Proving \(\angle3\cong\angle6\)

Using the Same - Side Interior Angles Postulate: \(\angle3\) is supplementary to \(\angle5\) (\(\angle3+\angle5 = 180^{\circ}\)). \(\angle6\) and \(\angle5\) are supplementary because they form a linear pair. Since \(\angle3\) and \(\angle6\) are supplementary to the same angle (\(\angle5\)), they are congruent.

Answer:

1. B (\(\angle4\)), D (\(\angle6\))
2. A (\(\angle1\)), C (\(\angle3\)), E (\(\angle7\))
3. \(\angle5\); \(\angle5\); supplementary; linear pair; supplementary