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Are these two triangles congruent? If so, explain why. Make sure to explain why something is congruent if not already given.

(1 point)
Responses

ASA: The two triangles are sharing a side, so those sides are congruent by the Reflexive Property of Congruence
ASA: The two triangles are sharing a side, so those sides are congruent by the Reflexive Property of Congruence

ASA: The perpendicular angles are both 90 degrees which means they are congruent.
ASA: The perpendicular angles are both 90 degrees which means they are congruent.

SAS: The perpendicular angles are both 90 degrees which means they are congruent.
SAS: The perpendicular angles are both 90 degrees which means they are congruent.

SAS: The two triangles are sharing a side, so those sides are congruent by the Reflexive Property of Congruence

1 answer

To determine if the two triangles are congruent, we need to analyze the conditions provided.

The congruence criteria require at least one of the following sets of conditions to prove triangle congruence:

  1. Side-Side-Side (SSS) – All three sides of one triangle are congruent to the three sides of another triangle.
  2. Side-Angle-Side (SAS) – Two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.
  3. Angle-Side-Angle (ASA) – Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.
  4. Angle-Angle-Side (AAS) – Two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle.
  5. Hypotenuse-Leg (HL) – In right triangles, the hypotenuse and one leg are congruent.

In your analysis:

  1. Reflexive Property of Congruence states that a side shared by both triangles is congruent to itself, which is a valid argument for proving congruence, particularly in the context of ASA or SAS.

  2. Right Angles: Both triangles having 90-degree angles means those angles are congruent.

Conclusion:

  • If both triangles share a side (let's call it side AB) and have two right angles (angles A and C), then we can conclude that the triangles are congruent by the ASA or SAS criteria:
    • For ASA, if you have two angles (the two right angles) and the included side (side AB which is shared), this satisfies the ASA condition.
    • For SAS, if you have two sides (AB and one other side in each triangle) and the included angle (the right angle formed by both sides), then this satisfies the SAS condition.

Thus, based on your reasoning, the two triangles can be classified as congruent via ASA, as the sides and angles satisfy this condition. Your earlier response mentioning the Reflexive Property of Congruence in relation to that shared side is crucial for this conclusion.

So, the final conclusion would be: Yes, the two triangles are congruent by ASA (or SAS) due to equal angles and the shared side, confirming their congruence through the properties of geometry.