Is this statement always, sometimes, or never true?If ∠H and ∠J are acute angles in a scalene right triangle, then sin(H)=sin(J).alwayssometimesneverSubmit

To determine whether the statement "If ∠H and ∠J are acute angles in a scalene right triangle, then $\sin(H) = \sin(J)$" is always, sometimes, or never true, we need to analyze the properties of angles in a scalene right triangle.

1. Definitions:

   • A scalene triangle is a triangle where all three sides are of different lengths, meaning all angles are also of different measures.
   • A right triangle contains one angle that is exactly $90^\circ$, making the sum of the other two angles $90^\circ$.
   • An acute angle is an angle that is less than $90^\circ$.

2. Analysis of Angles in a Scalene Right Triangle:

   • In a scalene right triangle, let’s denote the right angle as $90^\circ$ and the two acute angles as $H$ and $J$.
   • Since $H + J = 90^\circ$ and both $H$ and $J$ are acute, they cannot be equal; if one angle increases, the other must decrease to maintain the sum of $90^\circ$.

3. Sinusoidal Relationships:

   • The sine function is defined as: $\sin(H) = \frac{\text{opposite}}{\text{hypotenuse}} \quad \text{and} \quad \sin(J) = \frac{\text{opposite'} }{\text{hypotenuse}}$
   • Since $H$ and $J$ are different angles in a scalene triangle with distinct opposite sides, $\sin(H)$ cannot equal $\sin(J)$.

4. Conclusion:

   • Since $H$ and $J$ must be different in a scalene right triangle, their sine values must also be different.

Final Answer

The statement is never true.