The ASA Congruence Theorem Quick Check

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Four triangles are labeled upper A upper B upper C, upper E upper D upper C, upper A prime upper B prime upper C, and upper A double prime upper B double prime upper C, with the point upper C common to all three.

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?

(1 point)
Responses solve the answer carefully and tell me why

Rotate △ABC 90 degrees (counterclockwise) about vertex C, and then reflect △A′B′C′ across CA′¯¯¯¯¯¯¯¯¯.
Rotate triangle upper A upper B upper C 90 degrees (counterclockwise) about vertex upper C , and then reflect triangle upper A prime upper B prime upper C prime across Modifying above upper C upper A prime with bar .

Rotate △ABC −90 degrees (clockwise) about vertex C, and then reflect △A′B′C′ across CA′¯¯¯¯¯¯¯¯¯.
Rotate triangle upper A upper B upper C negative 90 degrees (clockwise) about vertex upper C , and then reflect triangle upper A prime upper B prime upper C prime across Modifying above upper C upper A prime with bar .

Rotate △ABC 360 degrees (counterclockwise), and then reflect △A′B′C′ across B′C¯¯¯¯¯¯¯¯¯.
Rotate triangle upper A upper B upper C 360 degrees (counterclockwise), and then reflect triangle upper A prime upper B prime upper C prime across Modifying above upper B prime upper C with bar .

Rotate △ABC −90 degrees (clockwise) about vertex C, and then reflect △A′B′C′ across B′C¯¯¯¯¯¯¯¯¯.
Rotate triangle upper A upper B upper C negative 90 degrees (clockwise) about vertex upper C , and then reflect triangle upper A prime upper B prime upper C prime across Modifying above upper B prime upper C with bar .
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Answer 1

To prove that triangles \( \triangle ABC \) and \( \triangle A''B''C'' \) are congruent using the ASA (Angle-Side-Angle) criterion, we need to demonstrate that angle \( A \) is equal to angle \( A'' \), side \( AC \) is equal to side \( A''C'' \), and angle \( C \) is equal to angle \( C'' \).

The ASA criterion states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.

To show how the ASA criterion follows from the definition of congruence in terms of rigid motions, we generally need to perform transformations that preserve distances and angles, such as rotations, reflections, and translations.

Let's analyze the options provided:

1. Rotate \( \triangle ABC \) 90 degrees (counterclockwise) about vertex \( C \), and then reflect \( \triangle A'B'C' \) across line \( CA' \).

2. Rotate \( \triangle ABC \) -90 degrees (clockwise) about vertex \( C \), and then reflect \( \triangle A'B'C' \) across line \( CA' \).

3. Rotate \( \triangle ABC \) 360 degrees (counterclockwise), and then reflect \( \triangle A'B'C' \) across line \( B'C' \).

4. Rotate \( \triangle ABC \) -90 degrees (clockwise) about vertex \( C \), and then reflect \( \triangle A'B'C' \) across line \( B'C' \).

Reasoning for Each Option

1. Counterclockwise 90-degree rotation + reflection over \( CA' \) does not necessarily match the position of triangle \( A''B''C'' \). This transformation will not align the corresponding angles and sides correctly.

2. Clockwise -90-degree rotation + reflection over \( CA' \) is likely more relevant, as it could maintain the position relative to point \( C \), but also requires verification if the sides and angles align properly.

3. 360-degree rotation just returns the triangle to its initial position; the subsequent reflection would not assist in establishing congruence with \( \triangle A''B''C'' \).

4. Clockwise -90-degree rotation + reflection across line \( B'C' \) seems to perform a transformation that might misalign the corresponding angles and sides and does not utilize the given angles and sides effectively to maintain congruence.

Conclusion

The option that best illustrates the ASA criterion through rigid motions transformations that align angles and sides is:

Rotate \( \triangle ABC \) -90 degrees (clockwise) about vertex \( C\), and then reflect \( \triangle A'B'C' \) across line \( CA' \).

This transformation keeps angle \( C \) fixed (necessary for congruence) and appropriately positions the first triangle relative to the second, thereby achieving the necessary conditions for ASA congruence.