Question

developing proof supply the missing reasons in the two - column proof. given: (aparallel b), (cparallel d) prove: (angle1) and (angle4) are supplementary. statements: 1) (angle1congangle2) 2) (cparallel d) 3) (angle2) and (angle3) are supplementary. 4) (aparallel b) 5) (angle3congangle4) 6) (angle1) and (angle4) are supplementary. reasons: 1) ? 2) given 3) ? 4) given 5) ? 6) ? drag and drop the correct choice into the drop areas on the image above.

Explanation:

Step1: Identify vertical - angle property

Vertical angles are congruent. Since $\angle1$ and $\angle2$ are vertical angles, $\angle1\cong\angle2$ because vertical angles are congruent.

Step2: Use given parallel - lines property

Given $c\parallel d$, when two parallel lines are cut by a transversal, corresponding angles are congruent. But for $\angle2$ and $\angle3$ being supplementary, we use the property that when two parallel lines ($c$ and $d$) are cut by a transversal, same - side interior angles are supplementary.

Step3: Use given parallel - lines property

Given $a\parallel b$, when two parallel lines are cut by a transversal, corresponding angles are congruent.

Step4: Use angle - congruence and substitution

If $\angle3\cong\angle4$ (from previous steps and parallel - line properties), and we know the relationships between other angles, we can prove the final statement.

Step5: Use angle - addition and supplementary - angle properties

Since $\angle1\cong\angle2$, and considering the relationships of $\angle2$ with $\angle3$ and $\angle3$ with $\angle4$ due to parallel lines, we can show that $\angle1$ and $\angle4$ are supplementary.

Answer:

1. Vertical angles are congruent.
2. Same - side interior angles of parallel lines ($c\parallel d$) are supplementary.
3. Given ($a\parallel b$).
4. Given.
5. Corresponding angles of parallel lines ($a\parallel b$) are congruent.
6. Angle addition and substitution based on previous angle - congruence and supplementary relationships.