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The HA theorem is a special case of the .

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Problem

The HA theorem is a special case of the ____.

Answer

AAS theorem

Step-by-step solution

  1. Understand the HA Theorem: The HA (Hypotenuse-Angle) theorem is a congruence postulate for right triangles. It states that if the hypotenuse and one acute angle of a right triangle are congruent to the hypotenuse and one acute angle of another right triangle, then the triangles are congruent.
  1. Relate HA to General Triangle Congruence Postulates: The general postulates for triangle congruence are SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), and AAS (Angle-Angle-Side).
  1. Analyze the HA Theorem's Conditions: The HA theorem involves two angles (the right angle and one acute angle) and the hypotenuse (a side). This means it's related to angle and side relationships.
  1. Consider the AAS Postulate: The AAS postulate states that if two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle, then the triangles are congruent.
  1. Connect HA to AAS: In a right triangle, we always have a right angle (90 degrees). If we know the hypotenuse and one acute angle (let's call it angle A), then we can find the third angle using the fact that the sum of angles in a triangle is 180 degrees. So, the third angle would be $180^\circ - 90^\circ - A = 90^\circ - A$. This means that if we know the hypotenuse and one acute angle of a right triangle, we implicitly know both acute angles and the hypotenuse. This fits the criteria of the AAS postulate (two angles and a non-included side) because the hypotenuse is a side, and we have two angles (the right angle and the given acute angle).

6. Evaluate Other Options:
* ASA Postulate: Requires two angles and the included side. HA provides a side (hypotenuse) and two angles, but the hypotenuse is not necessarily included between the two angles.
* SAS Postulate: Requires two sides and the included angle. HA does not directly provide two sides.
* SSS Postulate: Requires three sides. HA does not directly provide three sides.

  1. Conclusion: The HA theorem is a special case of the AAS postulate applied to right triangles.