The ASA Congruence Theorem Quick Check

Question

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Question
Which of the following pairs of triangles could be proven congruent by the ASA Congruence Theorem?(1 point)
Responses

Two congruent triangles, upper X upper Y upper Z and upper A upper B upper C are shown within an 11 column by 5 row grid.
Image with alt text: Two congruent triangles, upper X upper Y upper Z and upper A upper B upper C are shown within an 11 column by 5 row grid.

Two congruent triangles, upper A upper B upper C and upper Q upper R upper S are shown within an 11 column by 5 row grid. The sides upper A upper C and upper Q upper S are marked congruent by one tick mark.
Image with alt text: Two congruent triangles, upper A upper B upper C and upper Q upper R upper S are shown within an 11 column by 5 row grid. The sides upper A upper C and upper Q upper S are marked congruent by one tick mark.

Two congruent triangles, upper X upper Y upper Z and upper A upper B upper C are shown within an 11 column by 5 row grid.
Image with alt text: Two congruent triangles, upper X upper Y upper Z and upper A upper B upper C are shown within an 11 column by 5 row grid.

Two congruent triangles, upper A upper B upper C and upper Q upper R upper S are shown within an 11 columns by 5 row grid.
Image with alt text: Two congruent triangles, upper A upper B upper C and upper Q upper R upper S are shown within an 11 columns by 5 row grid.
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page 17 of 17

Answer 1

To use the ASA (Angle-Side-Angle) Congruence Theorem, you need to have two angles and the included side of one triangle congruent to two angles and the included side of another triangle.

From your descriptions, it looks like we need to identify which pair of triangles meets this criterion:

1. Two congruent triangles \( XYZ \) and \( ABC \) (no details about angles or sides provided).
2. In the pair \( ABC \) and \( QRS \), the sides \( AC \) and \( QS \) are marked congruent, but information about angles is not given.
3. Again, two congruent triangles \( XYZ \) and \( ABC \) (same as the first).
4. Again, \( ABC \) and \( QRS \) with no additional information provided.

Without specific details regarding angles in each of the pairs of triangles, it is difficult to definitively conclude. However, based on the format of the responses, the pair of triangles that mentions the congruence of angles and the included side would generally be the ones that can be proven congruent using ASA.

If pairs 2 and 4 imply that you know the angles of triangles \( ABC \) and \( QRS \) are congruent as well, then that would be a valid application of ASA. It sounds like the first pair might not provide enough information, and neither does the second, unless specific angle information is given.

Thus, without further details about the angles in the triangles, one cannot determine definitively from the provided information which pair satisfies the ASA conditions. Could you please clarify any specific angle congruences along with the sides for a more accurate response?