Question

in the diagram, dg = 12, gf = 4, eh = 9, and hf = 3. to prove that △dfe ~ △gfh by the sas similarity theorem, it can be stated that (\frac{df}{gf} = \frac{ef}{hf}) and
○ ∠dfe is 4 times greater than ∠gfh.
○ ∠fhg is (\frac{1}{4}) the measure of ∠fed.
○ ∠dfe is congruent to ∠gfh.
○ ∠fhg is congruent to ∠efd.

Explanation:

Step1: Recall SAS Similarity Theorem

The SAS (Side - Angle - Side) similarity theorem states that if two sides of one triangle are proportional to two sides of another triangle and the included angle between those sides is congruent, then the two triangles are similar.

Step2: Analyze the sides and included angle

We are given that \(\frac{DF}{GF}=\frac{EF}{HF}\) (from the problem statement, we can calculate \(DF = DG+GF=12 + 4=16\) and \(EF=EH + HF=9+3 = 12\), so \(\frac{DF}{GF}=\frac{16}{4}=4\) and \(\frac{EF}{HF}=\frac{12}{3}=4\), so the sides are proportional). For the SAS similarity, the included angle between the sides \(DF\) and \(EF\) in \(\triangle DFE\) and the included angle between the sides \(GF\) and \(HF\) in \(\triangle GFH\) should be congruent. The included angle for \(\triangle DFE\) between \(DF\) and \(EF\) is \(\angle DFE\), and the included angle for \(\triangle GFH\) between \(GF\) and \(HF\) is \(\angle GFH\). So \(\angle DFE\) must be congruent to \(\angle GFH\).

Let's analyze the other options:

• Option 1: Saying \(\angle DFE\) is 4 times greater than \(\angle GFH\) is wrong because for SAS similarity, the included angles must be congruent, not in a multiple relationship.
• Option 2: \(\angle FHG\) and \(\angle FED\) are not the included angles for the SAS similarity condition, so this is incorrect.
• Option 4: \(\angle FHG\) and \(\angle EFD\) are not the included angles for the SAS similarity condition, so this is incorrect.

Answer:

\(\angle DFE\) is congruent to \(\angle GFH\) (the third option)