Question

in the figure, line ( l ) divides the side lengths of ( \triangle 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} square \frac{ad}{ad} = \frac{ec}{ae} square \frac{ae}{ae} )	property of equality
( \frac{ab}{ad} = \frac{ac}{ae} )	substitution
( angle a cong angle a )	reflexive property
( \triangle abc square \triangle ade )	side - angle - side ______
( angle abc square angle ade )	corresponding angles of ____ triangles are ____.
line ( l ) is parallel to ( overline{bc} )	corresponding angles theorem

options: ( + ), ( \times ), ( - ), ( cong ), addition, congruence, multiplication, similarity, congruent, similar

Explanation:

Step1: Analyze the first blank in the property of equality step

We know that \(\frac{DB}{AD}=\frac{EC}{AE}\). To get \(\frac{AB}{AD}=\frac{AC}{AE}\), we need to add 1 to both sides of the given equation. Since \(\frac{DB}{AD}+ \frac{AD}{AD}=\frac{DB + AD}{AD}=\frac{AB}{AD}\) and \(\frac{EC}{AE}+\frac{AE}{AE}=\frac{EC + AE}{AE}=\frac{AC}{AE}\), the operation here is addition. So the first blank (operation) is \(+\) and the second blank (operation name) is "Addition".

Step2: Analyze the triangle similarity/congruence step

We have \(\frac{AB}{AD}=\frac{AC}{AE}\) and \(\angle A\cong\angle A\). By the Side - Angle - Side (SAS) similarity criterion (since the sides are in proportion and the included angle is equal), \(\triangle ABC\sim\triangle ADE\). So the first blank for the triangle relation is \(\sim\) (similar) and the second blank for the SAS type is "Similarity".

Step3: Analyze the corresponding angles step

Since \(\triangle ABC\sim\triangle ADE\), the corresponding angles of similar triangles are congruent. So \(\angle ABC\cong\angle ADE\), the first blank for angle relation is \(\cong\), the second blank for triangle type is "similar", and the third blank for angle relation is "congruent".

Answer:

• For the first row of blanks (Property of Equality): \(+\) (first blank), Addition (second blank)
• For the triangle row: \(\sim\) (first blank), Similarity (second blank)
• For the angle row: \(\cong\) (first blank), similar (second blank), congruent (third blank)

Filling the table:

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