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 { d f } { g f } = \frac { e f } { h f } \\) 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 the 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 angles are congruent, then the two triangles are similar.

Step2: Analyze the given ratios

We are given that \(\frac{DF}{GF}=\frac{DG + GF}{GF}=\frac{12 + 4}{4}=\frac{16}{4} = 4\) and \(\frac{EF}{HF}=\frac{EH+HF}{HF}=\frac{9 + 3}{3}=\frac{12}{3}=4\). So \(\frac{DF}{GF}=\frac{EF}{HF}\). Now we need to check the included angles.

Step3: Check the included angles

\(\angle DFE\) and \(\angle GFH\) are the included angles for the sides \(\frac{DF}{GF}\) and \(\frac{EF}{HF}\). For the SAS similarity theorem, we need \(\angle DFE\cong\angle GFH\)

Answer:

\(\angle DFE\) is congruent to \(\angle GFH\)