Question

In the diagram, .

Triangles G E F and J H I are shown. The length of side G F is 20 and the length of side I J is 10. Th elength of side F E is 40 and the length of side I H is 20.

To prove that the triangles are similar by the SAS similarity theorem, it needs to be proven that

J measures 60°.
J measures 30°.
I measures 60°.
I measures 30°.

Answer 1

To use the SAS (Side-Angle-Side) similarity theorem to prove that triangles \( GEF \) and \( JHI \) are similar, we need to establish that:

1. The lengths of two sides of one triangle are proportional to the lengths of two sides of the other triangle.
2. The included angle between those two sides in both triangles is equal.

From the information given:

- Length of side \( GF = 20 \)
- Length of side \( IJ = 10 \)
- Length of side \( FE = 40 \)
- Length of side \( IH = 20 \)

### Step 1: Check the proportions of the sides

We can calculate the ratios of the lengths of the corresponding sides:

\[
\frac{GF}{IJ} = \frac{20}{10} = 2
\]

\[
\frac{FE}{IH} = \frac{40}{20} = 2
\]

Both ratios are equal, which demonstrates that the sides are proportional:

\[
\frac{GF}{IJ} = \frac{FE}{IH} = 2
\]

### Step 2: Establishing the included angle

Now we need to show that the included angles \( \angle GEF \) and \( \angle JHI \) are equal. To do that, we need to identify or state what \( J \) represents in the context of angles related to \( \triangle JHI \).

Let's denote:

- \( \angle GEF \) as the angle between sides \( GF \) and \( FE \) (i.e., the angle opposite side \( GE \)).
- \( \angle JHI \) as the angle between sides \( IJ \) and \( IH \) (i.e., the angle opposite side \( JH \)).

Now, if it is given or established that \( J \) measures either \( 60^\circ \) or \( 30^\circ \) specifically and corresponds directly to the angles mentioned, we would apply that knowledge.

### Conclusion

Assuming the context implies \( J \) angles = \( 60^\circ \) or \( 30^\circ \) and matches \( \angle GEF \), we would denote that:

If \( \angle GEF = \angle JHI \), then:
- If \( J \) measures \( 60^\circ \), you could assert that both angles are equal, hence proving similarity via the angle condition of SAS.
- If \( J \) measures \( 30^\circ \) and corresponds as equal to \( \angle GEF \), it would prove similarity that way too.

Therefore, if any of the conditions above holds true, \( \triangle GEF \sim \triangle JHI \) can be concluded using the SAS similarity theorem based on the appropriate angle measure confirming equality.