Final answer:
The correct proportion demonstrating that the corresponding sides of the similar triangles are proportional, by the SAS similarity postulate, is option 2: 16 over 12 equals 12 over 9, which simplifies to 4 over 3 equals 4 over 3.
Step-by-step explanation:
The question is asking which proportion correctly shows that the corresponding sides of two similar triangles are proportional, assuming that the triangles are similar by the SAS similarity postulate. In similar triangles, the corresponding sides are in the same ratio. Therefore, both proportions involving corresponding sides must be equivalent for the triangles to be similar by SAS, which requires two sides to be proportional and the included angle to be congruent.
To find the correct proportion, we compare the ratios of the corresponding sides. Option 1) 16 over 12 equals 9 over 12 is incorrect because it simplifies to 4 over 3 equals 3 over 4, which are not equal. Option 2) 16 over 12 equals 12 over 9 simplifies to 4 over 3 equals 4 over 3, which are equal, so this proportion is correct. Option 3) 12 over 12 equals 16 over 9 simplifies to 1 equals 16 over 9, which is not true. Lastly, Option 4) 12 over 16 equals 16 over 9 is incorrect because it simplifies to 3 over 4 equals 16 over 9, and these are not equal. Therefore, the correct proportion is option 2: 16 over 12 equals 12 over 9, indicating that the corresponding sides of the triangles are proportional.