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 which is the true correct answer and 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 determine which option demonstrates that the triangles \( \triangle ABC \) and \( \triangle A''B''C'' \) are congruent by the ASA (Angle-Side-Angle) theorem using rigid motions, we need to analyze what the ASA criterion states and how rigid motions can be applied.

The ASA criterion implies 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. The rigid motions allowed (rotations and reflections) can be used to align the triangles and show that one triangle can be transformed into the other.

Let's analyze each of the given options:

1. Rotate \( \triangle ABC \) 90 degrees (counterclockwise) about vertex C, and then reflect \( \triangle A'B'C' \) across \( \overline{CA'} \).
2. Rotate \( \triangle ABC \) 90 degrees (counterclockwise) about vertex C, and then reflect \( \triangle A'B'C' \) across \( \overline{CA'} \).
3. Rotate \( \triangle ABC \) -90 degrees (clockwise) about vertex C, and then reflect \( \triangle A'B'C' \) across \( \overline{CA'} \).
4. Rotate \( \triangle ABC \) -90 degrees (clockwise) about vertex C, and then reflect \( \triangle A'B'C' \) across \( \overline{B'C} \).
5. Rotate \( \triangle ABC \) 360 degrees (counterclockwise), and then reflect \( \triangle A'B'C' \) across \( \overline{B'C} \).

Considering these options, we need to align triangle \( ABC \) with triangle \( A''B''C'' \) by positioning a pair of angles and the included side correctly:

• Options with a 90 degrees or -90 degrees rotation about point C will likely help in aligning common angles.
• Reflection across a line that connects matching points post-rotation is critical to fulfilling the ASA condition.

Option 3 seems most promising:

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

This choice effectively makes use of both a rotation and a reflection to match angles and reposition sides accordingly, satisfying the ASA condition.

Therefore, the correct answer is:

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

This method aligns the angles and included side perfectly, demonstrating congruence through rigid motions.