Question

in the figure, line l divides the side lengths of △abc proportionally such that $\frac{db}{ad}=\frac{ec}{ae}$. move options to the proof to show that line l is parallel to $overline{bc}$. statements reasons $\frac{db}{ad}=\frac{ec}{ae}$ given $\frac{db}{ad}square\frac{ad}{ad}=\frac{ec}{ae}square\frac{ae}{ae}$ property of equality $\frac{ab}{ad}=\frac{ac}{ae}$ substitution $angle acongangle a$ reflexive property △abc$square$△ade side - angle - side $angle abcsquareangle ade$ corresponding angles of $square$ triangles are $square$ line l is parallel to $overline{bc}$ corresponding angles theorem

Explanation:

Step1: Apply addition property of equality

$\frac{DB}{AD}+ 1=\frac{EC}{AE}+1$. Since $1 = \frac{AD}{AD}=\frac{AE}{AE}$, we get $\frac{DB + AD}{AD}=\frac{EC + AE}{AE}$, and $DB + AD=AB$, $EC + AE = AC$, so $\frac{AB}{AD}=\frac{AC}{AE}$.

Step2: Use similarity - criterion

For $\triangle ABC$ and $\triangle ADE$, we have $\frac{AB}{AD}=\frac{AC}{AE}$ and $\angle A\cong\angle A$ (reflexive property). By the Side - Angle - Side (SAS) similarity criterion, $\triangle ABC\sim\triangle ADE$.

Step3: Analyze corresponding angles

Since $\triangle ABC\sim\triangle ADE$, corresponding angles of similar triangles are congruent. So $\angle ABC\cong\angle ADE$.

Step4: Apply parallel - line theorem

By the Corresponding Angles Theorem, if corresponding angles are congruent, then the lines are parallel. So line $l$ is parallel to $\overline{BC}$.

Answer:

1. $+$, $+$; Addition Property of Equality
2. $\sim$; Similarity Criterion
3. $\cong$; similar; congruent