Consider the triangles.

Triangle G H I. Side G H is 16 inches, H I is 15 inches, G I is 10 inches. Angle G is 65 degrees, H is 48 degrees, I is 67 degrees. Triangle D E F. Side D E is 3 inches, E F is 3.2 inches, D F is 2 inches. Angle D is 67 degrees, E is 48 degrees, F is 65 degrees.

What can be concluded about these triangles? Check all that apply.
The corresponding angles are proportional.
The ratios of the corresponding sides are equivalent.
The corresponding sides are congruent.
The corresponding angles have the same measure.
DE corresponds to IH.

To determine what can be concluded about triangles \( \triangle GHI \) and \( \triangle DEF \), we need to compare their corresponding sides and angles. Let's review the given information:

- **Triangle \( \triangle GHI \)**:
  - Sides: \( GH = 16 \) inches, \( HI = 15 \) inches, \( GI = 10 \) inches
  - Angles: \( \angle G = 65^\circ \), \( \angle H = 48^\circ \), \( \angle I = 67^\circ \)

- **Triangle \( \triangle DEF \)**:
  - Sides: \( DE = 3 \) inches, \( EF = 3.2 \) inches, \( DF = 2 \) inches
  - Angles: \( \angle D = 67^\circ \), \( \angle E = 48^\circ \), \( \angle F = 65^\circ \)

### Analyzing Corresponding Angles

- \( \angle G = 65^\circ \) corresponds to \( \angle F = 65^\circ \)
- \( \angle H = 48^\circ \) corresponds to \( \angle E = 48^\circ \)
- \( \angle I = 67^\circ \) corresponds to \( \angle D = 67^\circ \)

Since the corresponding angles in both triangles are equal, the triangles are similar by the Angle-Angle-Angle (AAA) criterion. This implies that all corresponding angles have the same measure.

### Analyzing Corresponding Sides

To determine if the ratios of the corresponding sides are equivalent, we compare the ratios:

- \( GH : DE = 16 : 3 \)
- \( HI : EF = 15 : 3.2 \)
- \( GI : DF = 10 : 2 \)

Calculating these ratios:

- \( \frac{GH}{DE} = \frac{16}{3} \approx 5.33 \)
- \( \frac{HI}{EF} = \frac{15}{3.2} \approx 4.69 \)
- \( \frac{GI}{DF} = \frac{10}{2} = 5 \)

These ratios are not equivalent. Therefore, the ratios of the corresponding sides are not equivalent.

### Determining Congruency and Correspondence

Since the triangles are similar but not congruent (their corresponding sides are not equal or proportional), we can conclude the following:

1. The corresponding angles have the same measure.
2. The corresponding sides are not congruent.
3. The ratios of the corresponding sides are not equivalent.

Finally, examining if \( DE \) corresponds to \( IH \):

- \( DE \) corresponds to \( GH \) by comparing sides opposite the 48-degree angle.
- \( HI \) corresponds to \( EF \) by comparing sides opposite the 67-degree angle.

Therefore, \( DE \) does not correspond to \( IH \).

### Conclusion

From the given options, the correct conclusions are:

- The corresponding angles have the same measure.

**Incorrect conclusions are:**
- The corresponding angles are proportional. (Incorrect, angles are equal, not proportional)
- The ratios of the corresponding sides are equivalent. (Incorrect, as calculated)
- The corresponding sides are congruent. (Incorrect, sides are not equal)
- DE corresponds to IH. (Incorrect, \( DE \) corresponds to \( GH \))

Do you want any details or have any questions?

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**Relative Questions:**

1. How can we use the AAA criterion to determine if triangles are similar?
2. What is the difference between similar and congruent triangles?
3. How do you find the ratio of corresponding sides in similar triangles?
4. How can we prove that two triangles are similar using their angles?
5. What is the significance of proportional sides in similar triangles?
6. How do you determine the corresponding sides in two triangles?
7. What methods can be used to verify the similarity of triangles?
8. Can two triangles be similar if they have one pair of corresponding angles equal?

**Tip:** When comparing triangles, always ensure to correctly identify and match corresponding sides and angles for accurate conclusions.

Explore the similarity between triangles GHI and DEF based on their corresponding angles and sides. Understand how to apply the Angle-Angle-Angle (AAA) criterion to determine triangle similarity. Suitable for high school students.

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