Final answer:
Transformations such as reflections across the axes, translations, and rotations preserve the congruency of figures by altering their positions without changing their shapes or sizes. However, dilations are not congruence transformations as they change the size of the figure.
Step-by-step explanation:
Transformations that result in a figure being congruent to the given figure are operations that change the figure’s position without changing its size or shape. Based on the options provided:
• I. A reflection across the x-axis.
• II. A reflection across the y-axis.
• III. A translation to the right 2 and up 3.
• IV. A rotation of 180 degrees counterclockwise.
• V. A dilation by a factor of 2.
Options I, II, III, and IV are transformations that preserve congruency as they do not alter the size of the figure—only its position or orientation. Reflections across the x-axis or y-axis, translations, and rotations simply move or flip the figure without changing its dimensions. However, option V, a dilation by a factor of 2, would result in a figure that is not congruent to the original because it changes the size of the figure, effectively doubling all lengths, and thereby altering proportions while maintaining shape similarity. Therefore, the transformations that will result in congruent figures to the original are reflections, translations, and rotations. In contrast, a dilation changes the size and does not preserve congruency.