Question

given \\(\overline{ab} \parallel \overline{dc}\\), complete the flowchart proof below. note that the last statement and reason have both been filled in for you.

diagram of triangles cde and abe with e as the intersection point, ab parallel to dc, and some sides marked congruent

for each box, choose a statement format from the dropdown menu. you will then be able to change the letters to match the diagram for this problem.

flowchart with boxes for statements and reasons, leading to \\(\triangle abe \cong \triangle cde\\) with reason sas

Explanation:

Step1: Vertical angles are congruent

$\angle AEB\cong\angle DEC$ (Vertical angles are formed when two lines intersect. By the vertical - angles theorem, $\angle AEB$ and $\angle DEC$ are congruent)

Step2: Alternate - interior angles are congruent

Since $\overline{AB}\parallel\overline{DC}$, by the alternate - interior angles theorem, $\angle A\cong\angle D$ (When two parallel lines are cut by a transversal, alternate - interior angles are congruent. Here, the transversal is $\overline{AD}$)

Step3: Given side - equality

$\overline{AE}\cong\overline{DE}$ and $\overline{BE}\cong\overline{CE}$ (These are given as marked on the diagram)

Answer:

The first box (top - left): $\angle AEB\cong\angle DEC$, Reason: Vertical Angles Theorem;
The second box (left - middle): $\angle A\cong\angle D$, Reason: Alternate - Interior Angles Theorem (because $\overline{AB}\parallel\overline{DC}$);
The third box (right - middle): $\overline{AE}\cong\overline{DE}$ and $\overline{BE}\cong\overline{CE}$, Reason: Given (marked on the diagram)