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 (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 −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 . 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 (counterclockwise) about vertex C, and then

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GPT-4o mini · Answer

To prove that triangle $ \triangle ABC $ is congruent to triangle $ \triangle A''B''C'' $ using the Angle-Side-Angle (ASA) criterion and the definition of congruence in terms of rigid motions, we can break down the process: 1. **Identify the Corresponding Parts**: We need to align the angles and the side of one triangle with those of the other triangle. 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 triangles are congruent. 2. **Applying Rigid Motions**: Rigid motions (rotations, reflections, and translations) preserve distances and angles. We can use these to show that one triangle can be transformed into the other. The correct approach from the options you've provided should include: - **A rotation about a vertex**: This helps position one angle to match the corresponding angle of the other triangle. - **A reflection across a side**: This aligns the triangles. Among the responses you listed, the options are somewhat similar in structure, but it is important to choose the one that properly applies the ASA principle. **Response that Fits ASA**: "Rotate $ \triangle ABC $ $ -90 $ degrees (clockwise) about vertex $ C $, and then reflect $ \triangle A'B'C' $ across $ CA' $." This response takes one angle of $ \triangle ABC $ and manipulates it so that it corresponds with the angle of $ \triangle A'B'C' $, which helps satisfy the ASA conditions. Therefore, effectively applying both rigid motions above will result in $ \triangle ABC $ being transformed into $ \triangle A''B''C'' $, proving their congruence.

:3 · Answer

sooo which one abc or d

GPT-4o mini · Answer

To determine which option correctly shows how the ASA criterion follows from the definition of congruence in terms of rigid motions, we need to look closely at the provided options.

Among the provided options, the one that best demonstrates the ASA criterion is:

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

This option (let’s assume it's option A) is suitable as it describes a rotation followed by a reflection, which is a common method used to show congruence between two triangles using angles and included sides.

So, if that corresponded to option A (the first one), then A is correct. If the options you provided initially had labels corresponding to certain letters (like A, B, C, D), please confirm which was which to clarify the exact answer.