- Analyze the given conditions for the two right triangles, noting that the right angle is non-included.
- Conclude that the given sides must be the hypotenuse and one leg.
- Evaluate the standard congruence theorems (SAS, SSS, AAS, HL) and identify that HL is the one that specifically addresses the hypotenuse and leg.
- Identify the correct theorem as \(\boxed{HL}\).
Explanation
- Analyze the given conditions.
The problem describes two right triangles with a specific congruence condition involving two sides and a "non-included" right angle. Let us define the two triangles as \(\triangle ABC\) and \(\triangle DEF\), where \(\angle B = \angle E = 90^\circ\). The condition states that we are given two sides of each triangle and the right angle, but the right angle is not the angle included between those two sides.
- Determine the nature of the given sides.
In a right triangle, the right angle (\(90^\circ\)) is formed by the two legs. If the given angle (the right angle) is not included between the two given sides, then the two sides cannot be the two legs. One of the sides must be the hypotenuse, and the other side must be one of the legs. This implies that for each triangle, we have a congruent hypotenuse and a congruent leg.
- Evaluate the congruence theorems.
- SAS (Side-Angle-Side) requires the angle to be included between the two sides. Since the problem explicitly states the right angle is non-included, SAS does not apply.
- SSS (Side-Side-Side) requires three sides, but we only have information about two sides.
- AAS (Angle-Angle-Side) requires two angles and one non-included side. While we know the right angles are congruent, we only have two sides, not two angles.
- HL (Hypotenuse-Leg) theorem states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and a corresponding leg of another right triangle, then the triangles are congruent. This matches our conclusion that we have a congruent hypotenuse and a leg.
- Conclusion.
Since the conditions given correspond to having a congruent hypotenuse and a leg in each right triangle, the appropriate congruence theorem is the HL (Hypotenuse-Leg) theorem.
The final answer is \(\boxed{HL}\).
Examples
Understanding the HL theorem is crucial in fields like civil engineering and architecture, where structural stability often relies on congruent triangular supports. For instance, if you are building two identical rectangular trusses for a bridge, you only need to ensure the hypotenuse and one leg of the triangles are of the same length to guarantee the entire structures are identical, making the manufacturing process much more efficient. This principle ensures that parts are interchangeable and that the construction meets safety standards with minimal testing.