https://file.aiquickdraw.com/mathbot/file/1727372951498pbzm3kjz.jpg

The problem is asking which sequence of rigid motions (rotations and reflections) will prove that the triangles \( \triangle ABC \cong \triangle A''B''C'' \) using the ASA (Angle-Side-Angle) criterion. Let's break down the diagram and the available options:

### Given Information:
1. The triangles are \( \triangle ABC \) and \( \triangle A''B''C'' \), and we need to show they are congruent by rigid motions.
2. The ASA criterion tells us that two triangles are congruent if two angles and the included side in one triangle are congruent to two angles and the included side in another triangle.

### Available Rigid Motions:
The problem provides various sequences of rigid motions (rotations and reflections) about specific points or sides. We need to identify which transformation maps \( \triangle ABC \) onto \( \triangle A''B''C'' \) properly, using the ASA criterion.

### Options Breakdown:
1. **Option 1**: Rotate \( \triangle ABC \) \(-90^\circ\) (clockwise) about vertex \( C \), and then reflect \( \triangle A'B'C' \) across \( \overline{CA'} \).
2. **Option 2**: Rotate \( \triangle ABC \) \(-90^\circ\) (clockwise) about vertex \( C \), and then reflect \( \triangle A'B'C' \) across \( \overline{BC} \).
3. **Option 3**: Rotate \( \triangle ABC \) \(90^\circ\) (counterclockwise) about vertex \( C \), and then reflect \( \triangle A'B'C' \) across \( \overline{CA'} \).
4. **Option 4**: Rotate \( \triangle ABC \) \(360^\circ\) (counterclockwise), and then reflect \( \triangle A'B'C' \) across \( \overline{BC} \).

### Analysis:
- **Rotation and Reflection**: For the triangles to map onto each other, a combination of rotation and reflection should align corresponding sides and angles. We are looking for a transformation that preserves the angle congruence (ASA) and properly aligns the sides.

- **Rotation about \( C \)**: Since \( C \) is a common vertex, the rotation should keep point \( C \) fixed while transforming points \( A \) and \( B \) appropriately. Given the structure of the problem, a \( -90^\circ \) (clockwise) rotation seems appropriate because it aligns the angles correctly.

- **Reflection**: The reflection should then map the triangle onto its final position, aligning \( A' \) and \( B' \) onto \( A'' \) and \( B'' \).

### Correct Answer:
The correct transformation sequence that proves congruence via rigid motions is **Option 1: Rotate \( \triangle ABC \) \(-90^\circ\) (clockwise) about vertex \( C \), and then reflect \( \triangle A'B'C' \) across \( \overline{CA'} \).**

Would you like further details on this reasoning? Let me know if you have any questions!

Here are some related questions:
1. How do rotations preserve angle congruence in transformations?
2. Why is reflection needed to finalize the transformation?
3. What would happen if a different rotation was applied instead of \( -90^\circ \)?
4. How does the ASA criterion help in proving triangle congruence?
5. Can rigid motions be used to prove congruence for any pair of triangles?

**Tip:** Always consider the relative positioning of vertices when determining how rigid motions should be applied for congruence.

To prove ∆ABC ≅ ∆A''B''C'', which of the following shows how the ASA criterion follows from the definition of congruence in terms of rigid motions?

Learn how to prove that ∆ABC ≅ ∆A''B''C'' using rigid motions and the ASA criterion. This problem explores key concepts in geometry, including triangle congruence, rotations, and reflections, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation