To determine if the two triangles are congruent, we need to analyze the statements made in your responses based on the criteria for triangle congruence.
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SSA (Side-Side-Angle): This is not a valid congruence theorem. Two triangles can sometimes have two sides and a non-included angle that are equal but still not be congruent.
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SAS (Side-Angle-Side): You mention that the two triangles share a side, and the angles are both 90 degrees. If this is the case, the shared side (let's call it "AB") is congruent to itself (by the Reflexive Property), and if we can say that the sides adjacent to the right angles are also equal, then we can use the SAS theorem. However, we need to confirm that the angle between the two congruent sides is indeed one of the 90-degree angles.
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ASA (Angle-Side-Angle): This can also indicate congruence if we have two angles and the side between them that are congruent. Your mention of vertical angles being congruent supports this if the triangle setup allows for accomplishing this type of congruence.
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HL (Hypotenuse-Leg): This applies specifically to right triangles. If the hypotenuses are equal and one pair of legs is equal, then the triangles are congruent by HL.
In conclusion, if you verify that specific sides and angles are congruent, you may conclude that the triangles are congruent using SAS, ASA, or HL. Therefore:
- If the triangles are right triangles and you can establish the hypotenuses and at least one leg to be equal, then they are congruent by HL.
- If you established two angles and the included side as congruent, then they are congruent by ASA.
- If you showed a side and the angles adjacent to that side are congruent, then you could potentially use SAS.
If none of these conditions can be met, then the statement "these two triangles are not congruent" using SSA would be justified. Thus, it all depends on what information we can concretely establish as congruent.
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