Question

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

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 is congruent, then the two triangles are similar.

Step2: Analyze the given sides and angles

First, calculate the lengths of \(DF\) and \(EF\). We know that \(DG = 12\) and \(GF=4\), so \(DF=DG + GF=12 + 4=16\). Also, \(EH = 9\) and \(HF = 3\), so \(EF=EH+HF = 9 + 3=12\).

Now, check the ratios: \(\frac{DF}{GF}=\frac{16}{4} = 4\) and \(\frac{EF}{HF}=\frac{12}{3}=4\), so \(\frac{DF}{GF}=\frac{EF}{HF}\).

For the SAS similarity theorem, we need the included angle to be congruent. The included angle between \(DF\) and \(EF\) in \(\triangle DFE\) is \(\angle DFE\), and the included angle between \(GF\) and \(HF\) in \(\triangle GFH\) is \(\angle GFH\). So we need \(\angle DFE\cong\angle GFH\) (since they are the included angles for the proportional sides).

Let's analyze the other options:

• Option 1: Saying \(\angle DFE\) is 4 times greater than \(\angle GFH\) means they are not congruent, so this does not satisfy SAS similarity.
• Option 2: \(\angle FHG\) and \(\angle EFD\) are not the included angles for the proportional sides, so this is incorrect.
• Option 4: \(\angle FHG\) and \(\angle FED\) are not related to the included angles for the SAS similarity of \(\triangle DFE\) and \(\triangle GFH\).

Answer:

\(\angle DFE\) is congruent to \(\angle GFH\) (the third option in the list of angle statements).