https://file.aiquickdraw.com/mathbot/file/1726165439709qls0eorr.jpg

Let's break down the question step-by-step and solve it.

You are given two triangles, \( \triangle ABC \) and \( \triangle A'B'C' \), where Shawna used a reflection followed by a dilation to transform \( \triangle ABC \) to show that \( \triangle A'B'C' \). Based on the image:

- Triangle \( \triangle ABC \) is smaller and is reflected/dilated to form \( \triangle A'B'C' \).
- The angles in \( \triangle A'B'C' \) are provided: \( \angle B'C'B' = 110^\circ \) and \( \angle A'C'B' = 45^\circ \).

The options in the question include:
1. \( \angle A \cong \angle A' \)
2. \( \triangle ABC \sim \triangle A'B'C' \) (similarity)
3. \( \triangle ABC \cong \triangle A'B'C' \) (congruence)
4. \( m \angle B = 120^\circ \)

### Solving:

1. **Reflection and Dilation Properties:**
   - A reflection preserves angles but can change the orientation.
   - A dilation preserves the shape and the angles but alters the size.
   - Therefore, the triangles \( \triangle ABC \) and \( \triangle A'B'C' \) should be **similar**, not congruent, because they have the same angles but different sizes.

2. **Angle Correspondence:**
   - Since a reflection followed by a dilation preserves angles, \( \angle A \) should be congruent to \( \angle A' \). Therefore, **\( \angle A \cong \angle A' \)** is **true**.
   
3. **Similarity:**
   - Because the transformation includes a dilation (which changes size but preserves shape), \( \triangle ABC \sim \triangle A'B'C' \). Therefore, **\( \triangle ABC \sim \triangle A'B'C' \)** is **true**.
   
4. **Congruence:**
   - Congruent triangles must have the same size and shape, but since dilation changes the size, \( \triangle ABC \) and \( \triangle A'B'C' \) are not congruent. Therefore, **\( \triangle ABC \cong \triangle A'B'C' \)** is **false**.
   
5. **Angle \( \angle B \):**
   - The problem doesn't provide enough information to directly calculate \( m \angle B \) based on the image alone. However, since \( \triangle A'B'C' \) has a 110° angle and a 45° angle, the remaining angle must be \( 25^\circ \) (since the sum of angles in a triangle is 180°). The angle correspondence suggests that \( m \angle B = 25^\circ \), not 120°. Therefore, **\( m \angle B = 120^\circ \)** is **false**.

### Correct Answers:
- \( \angle A \cong \angle A' \)
- \( \triangle ABC \sim \triangle A'B'C' \)

Would you like further explanation on any of the steps? Here are five related questions for practice:

1. How can you prove two triangles are similar using transformations?
2. What properties of a triangle are preserved under a reflection?
3. How does a dilation affect the sides and angles of a triangle?
4. What are the conditions for two triangles to be congruent?
5. How would you calculate the missing angle in a triangle if you know two angles?

**Tip:** Always check whether the transformation changes the size of a figure (dilation) or just its position and orientation (reflection, rotation, translation).

Shawna used a reflection followed by a dilation to transform ΔABC to show ΔA'B'C' as shown. Which of the following statements are true?

Solve a geometry problem involving reflection and dilation of triangles. Understand the relationship between ΔABC and ΔA'B'C', exploring properties of similar and congruent triangles and angle measures.

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