Question

problem 23
given: $overline{be}paralleloverline{cd}$, $overline{be}congoverline{cd}$, $overline{ae}congoverline{ed}$
prove: $overline{ab}paralleloverline{ce}$

1. $angle acongangle ced$
2. corresponding parts of congruent tri - angles are congruent (c.p.c.t.c.)
3. $overline{ab}paralleloverline{ce}$
4. click here to insert

Explanation:

Step1: Identify angle relationship

We know that $\angle A\cong\angle CED$ from statement 4. These are alternate - interior angles for lines $\overline{AB}$ and $\overline{CE}$ with transversal $\overline{AE}$.

Step2: Apply parallel - line theorem

By the theorem that if two lines are cut by a transversal and the alternate - interior angles are congruent, then the two lines are parallel, since $\angle A$ and $\angle CED$ are alternate - interior angles and $\angle A\cong\angle CED$, we can conclude that $\overline{AB}\parallel\overline{CE}$.

Answer:

1. If two lines are cut by a transversal and the alternate - interior angles are congruent, then the lines are parallel.