Triangles A B C and Q R S are shown. Sides A B and Q R are congruent. Angles C A B and R Q S are congruent. Angles Q S R and A C B are congruent.
Two rigid transformations are used to map TriangleABC to TriangleQRS. The first is a translation of vertex B to vertex R. What is the second transformation?

a reflection across the line containing AB
a rotation about point B
a reflection across the line containing CB
a rotation about point C

The given problem involves mapping two triangles, \( \triangle ABC \) and \( \triangle QRS \), using two rigid transformations.

### Known Information:
- Sides \( AB \) and \( QR \) are congruent.
- Angles \( \angle CAB \) and \( \angle RQS \) are congruent.
- Angles \( \angle QSR \) and \( \angle ACB \) are congruent.
- The first transformation is a translation that maps vertex \( B \) to vertex \( R \).

### Analysis:
Once \( B \) is mapped to \( R \), the next step is to align the rest of the triangle.

- Since angles \( \angle CAB \) and \( \angle RQS \) are congruent, and angles \( \angle ACB \) and \( \angle QSR \) are congruent, this suggests that the two triangles share a similar orientation.
- After the translation, the triangle will still need to be rotated to align the remaining points.

### Best Match:
A **rotation about point \( B \)** will map the remaining points \( A \) to \( Q \) and \( C \) to \( S \), as the angles and sides match accordingly after the translation.

### Answer:
The second transformation is **a rotation about point \( B \)**.

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Do you need further details on this explanation, or have any questions? Here are some related questions for deeper exploration:

1. What defines congruent triangles in terms of transformations?
2. How do translations and rotations preserve the properties of shapes?
3. Why do rotations around a point maintain the distances between vertices?
4. How can reflections be used to map congruent figures?
5. What are the properties of rigid transformations in geometry?

**Tip**: When dealing with rigid transformations, remember that distances and angles are preserved, meaning congruent shapes can be mapped through translations, rotations, or reflections without altering their size or shape.

Triangles A B C and Q R S are shown. Sides A B and Q R are congruent. Angles C A B and R Q S are congruent. Angles Q S R and A C B are congruent. Two rigid transformations are used to map Triangle ABC to Triangle QRS. The first is a translation of vertex B to vertex R. What is the second transformation?
a reflection across the line containing AB
a rotation about point B
a reflection across the line containing CB
a rotation about point C

Discover how to map Triangle ABC to Triangle QRS using two rigid transformations. The problem explores congruent triangles, specifically involving a translation and a rotation about point B. Suitable for students in Grades 8-10, this solution demonstrates congruence and geometric transformations.

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