The ASA Congruence Theorem Quick Check

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Use the image to answer the question.

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

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 \( \triangle ABC \cong \triangle A''B''C'' \) using the ASA (Angle-Side-Angle) criterion through rigid motions, we need to show how we can manipulate triangle \( ABC \) so that it fits perfectly over triangle \( A''B''C'' \).

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.

Without specific measurements or angles given, I can determine the correct response by analyzing the transformations listed.

Here's the breakdown:

1. Rotation followed by a reflection is a standard way to manipulate triangles. The rotation should be enough to align the triangles based on their angles, and the reflection will help line up the sides correctly.

2. Since point \( C \) is common in all triangles, rotating around \( C \) is logical.

3. After the rotation, the reflection should occur across a line that connects appropriate vertices, ensuring that the triangles line up as needed.

Based on this reasoning, the correct choice among the provided options would likely be:

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

This reflects the necessary steps of aligning the triangles through a rotation followed by a reflection across a side that serves to align the other components of the triangles.

The answer, therefore, is:

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