Question

look at the image below. given: \\(\frac{ge}{de} = \frac{fe}{ce}\\) prove: \\(\triangle efg sim \triangle ecd\\) complete the proof by dragging the correct reason for each step of the proof to the corresponding box. statements reasons 1. \\(\frac{ge}{de} = \frac{fe}{ce}\\) 2. \\(angle gef cong angle dec\\) 3. \\(\triangle dec sim feg\\) substitution property, corresponding angles, reflexive property, given, sss similarity theorem, ssa similarity theorem, sas similarity theorem, asa similarity theorem

Explanation:

Step1: Reason for Statement 1

The first statement \(\frac{GE}{DE}=\frac{FE}{CE}\) is given in the problem, so the reason is "Given".

Step2: Reason for Statement 2

\(\angle GEF\) and \(\angle DEC\) are the same angle (they share the same vertex and sides), so by the Reflexive Property, \(\angle GEF\cong\angle DEC\).

Step3: Reason for Statement 3

We have two sides in proportion (\(\frac{GE}{DE}=\frac{FE}{CE}\)) and the included angle congruent (\(\angle GEF\cong\angle DEC\)), so by the SAS (Side - Angle - Side) Similarity Theorem, \(\triangle DEC\sim\triangle FEG\) (and since similarity is symmetric, \(\triangle EFG\sim\triangle ECD\) as well).

Answer:

1. For the first row (Statement 1: \(\frac{GE}{DE}=\frac{FE}{CE}\)): Given
2. For the second row (Statement 2: \(\angle GEF\cong\angle DEC\)): Reflexive Property
3. For the third row (Statement 3: \(\triangle DEC\sim\triangle FEG\)): SAS Similarity Theorem