In right triangles, congruent angles always correspond to which of the following properties?

• A. Equal ratios of corresponding sides, so their trigonometric function values are equal ($\sin$, $\cos$, $\tan$ etc.)
• B. Equal areas for the triangles
• C. Different hypotenuse lengths but equal trigonometric function values ($\sin$, $\cos$, $\tan$ etc.)
• D. Different ratios of corresponding sides

Answer: (A)

Verified step by step guidance

1. Recall that in right triangles, if two angles are congruent, the triangles are similar by the Angle-Angle (AA) similarity criterion.
2. Understand that similar triangles have corresponding sides in proportion, meaning the ratios of corresponding sides are equal.
3. Recognize that trigonometric functions like sine, cosine, and tangent are defined as ratios of sides in a right triangle (e.g., \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\)).
4. Since the triangles are similar, the ratios of corresponding sides are equal, so the values of the trigonometric functions for the congruent angles are also equal.
5. Conclude that congruent angles correspond to equal ratios of corresponding sides and equal trigonometric function values, not necessarily equal areas or hypotenuse lengths.