Question

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Explanation:

Step1: Add 1 to both sides

$\frac{DB}{AD} + 1 = \frac{EC}{AE} + 1$
$\frac{DB + AD}{AD} = \frac{EC + AE}{AE}$

Step2: Substitute for sum segments

$\frac{AB}{AD} = \frac{AC}{AE}$

Step3: Identify shared angle

$\angle A \cong \angle A$

Step4: Apply SAS similarity

$\triangle ABC \sim \triangle ADE$ (Side-Angle-Side Similarity)

Step5: State corresponding angles rule

$\angle ABC \cong \angle ADE$ (Corresponding angles of similar triangles are congruent)

Step6: Use parallel angle theorem

Line $l \parallel \overline{BC}$ (Corresponding Angles Theorem)

Answer:

Filled-in proof blanks (in order):

1. $\boldsymbol{+}$ (first blank), $\boldsymbol{+}$ (second blank); Reason: $\boldsymbol{Addition}$
2. $\boldsymbol{\sim}$; Reason: $\boldsymbol{Similarity}$
3. $\boldsymbol{\cong}$; Blanks: $\boldsymbol{similar}$, $\boldsymbol{congruent}$

Completed proof table:

Statements	Reasons
$\frac{DB}{AD} \boldsymbol{+} \frac{AD}{AD} = \frac{EC}{AE} \boldsymbol{+} \frac{AE}{AE}$	$\boldsymbol{Addition}$ Property of Equality
$\frac{AB}{AD} = \frac{AC}{AE}$	Substitution
$\angle A \cong \angle A$	Reflexive Property
$\triangle ABC \boldsymbol{\sim} \triangle ADE$	Side-Angle-Side $\boldsymbol{Similarity}$
$\angle ABC \boldsymbol{\cong} \angle ADE$	Corresponding angles of $\boldsymbol{similar}$ triangles are $\boldsymbol{congruent}$
Line $l$ is parallel to $\overline{BC}$	Corresponding Angles Theorem