Question
Are these two triangles congruent? If so, explain why. Make sure to explain why something is congruent if not already given
(1 point)
Responses
These 2 triangles are not congruent because we can not use SSA as a congruence theorem
These 2 triangles are not congruent because we can not use SSA as a congruence theorem
SAS: the two triangles share a side, which means that side is congruent to itself by the Reflexive Property of Congruence. The angles are congruent also because they are both 90 degrees.
SAS: the two triangles share a side, which means that side is congruent to itself by the Reflexive Property of Congruence. The angles are congruent also because they are both 90 degrees.
ASA: The angles are congruent by vertical angles theorem.
ASA: The angles are congruent by vertical angles theorem.
HL: the hypotenuses are the same in both triangles. Therefore, the hypotenuse are congruent to themselves by the Reflexive Property of Congruence. The angles are congruent also because they are both 90 degrees.
(1 point)
Responses
These 2 triangles are not congruent because we can not use SSA as a congruence theorem
These 2 triangles are not congruent because we can not use SSA as a congruence theorem
SAS: the two triangles share a side, which means that side is congruent to itself by the Reflexive Property of Congruence. The angles are congruent also because they are both 90 degrees.
SAS: the two triangles share a side, which means that side is congruent to itself by the Reflexive Property of Congruence. The angles are congruent also because they are both 90 degrees.
ASA: The angles are congruent by vertical angles theorem.
ASA: The angles are congruent by vertical angles theorem.
HL: the hypotenuses are the same in both triangles. Therefore, the hypotenuse are congruent to themselves by the Reflexive Property of Congruence. The angles are congruent also because they are both 90 degrees.
Answers
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Answered by
GPT-4o mini
The responses provided give various reasons for determining the congruence of two triangles, but they seem to reference different congruence criteria. Here’s a breakdown of each:
1. **SSA (Side-Side-Angle)**: This is not a valid congruence theorem. While you can have two sides and a non-included angle, this does not guarantee that the two triangles are congruent. Thus, if the reasoning was based solely on SSA, it would be correct to state that the two triangles are not congruent.
2. **SAS (Side-Angle-Side)**: This theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. If the two triangles share a side (let’s call it side AB) and have a right angle (which is 90 degrees) at the same vertex (say, angle A or B), then the shared side is congruent to itself by the Reflexive Property, and the right angles are congruent, providing enough information to declare that these two triangles can be congruent by this criterion.
3. **ASA (Angle-Side-Angle)**: This criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. If this is the reasoning proposed, and if the angles mentioned are indeed congruent (possibly through the Vertical Angles Theorem if sharing a vertex), then this could also justify congruence.
4. **HL (Hypotenuse-Leg)**: This is a specific criterion for right triangles. It states that two right triangles are congruent if the hypotenuse and one leg of one triangle are congruent to the hypotenuse and one leg of the other triangle. If the triangles in question are right triangles, and you can demonstrate that the hypotenuses and a leg are congruent, then you successfully establish congruence using this criterion.
In summary, to determine whether the triangles are congruent, you need to present a consistent criterion based on the given information (SSA, SAS, ASA, or HL). If you can establish that the conditions for SAS, ASA, or HL are met, then the triangles can be deemed congruent. However, if the reasoning is solely based on SSA without any of the other criteria being satisfied, you would conclude that they are not congruent.
1. **SSA (Side-Side-Angle)**: This is not a valid congruence theorem. While you can have two sides and a non-included angle, this does not guarantee that the two triangles are congruent. Thus, if the reasoning was based solely on SSA, it would be correct to state that the two triangles are not congruent.
2. **SAS (Side-Angle-Side)**: This theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. If the two triangles share a side (let’s call it side AB) and have a right angle (which is 90 degrees) at the same vertex (say, angle A or B), then the shared side is congruent to itself by the Reflexive Property, and the right angles are congruent, providing enough information to declare that these two triangles can be congruent by this criterion.
3. **ASA (Angle-Side-Angle)**: This criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. If this is the reasoning proposed, and if the angles mentioned are indeed congruent (possibly through the Vertical Angles Theorem if sharing a vertex), then this could also justify congruence.
4. **HL (Hypotenuse-Leg)**: This is a specific criterion for right triangles. It states that two right triangles are congruent if the hypotenuse and one leg of one triangle are congruent to the hypotenuse and one leg of the other triangle. If the triangles in question are right triangles, and you can demonstrate that the hypotenuses and a leg are congruent, then you successfully establish congruence using this criterion.
In summary, to determine whether the triangles are congruent, you need to present a consistent criterion based on the given information (SSA, SAS, ASA, or HL). If you can establish that the conditions for SAS, ASA, or HL are met, then the triangles can be deemed congruent. However, if the reasoning is solely based on SSA without any of the other criteria being satisfied, you would conclude that they are not congruent.
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