Final answer:
Triangles DEF and RPQ are congruent. DEF can be mapped to RPQ by reflecting it across the x-axis and translating it 2 units left, making the triangles align perfectly.
Step-by-step explanation:
Are triangles DEF and RPQ congruent? To determine the congruence, we must examine the possibility of mapping triangle DEF onto triangle RPQ using rigid transformations such as reflections, translations, or rotations. Examining the coordinates of the given points in each triangle, we can analyze the distances between the corresponding points to check for congruence.
Triangle DEF has vertices at (1, 0), (1, 3), and (5, 0), which means the lengths of the sides are 3 units vertically and 4 units horizontally, forming a 3-4-5 right triangle. Triangle RPQ has vertices at (-1, -2), (-1, -5), and (-5, -2), which has the side lengths of 3 units vertically and 4 units horizontally, also forming a 3-4-5 right triangle. In both triangles, the right angle is indicated by the vertices with equal x-values (1 for DEF and -1 for RPQ).
To map triangle DEF to triangle RPQ, we can reflect DEF across the x-axis, after which DEF will have vertices at (1, 0), (1, -3), and (5, 0). Then, by translating the triangle 2 units to the left, we will have the vertices at (-1, 0), (-1, -3), and (3, 0), which align perfectly with the vertices of triangle RPQ. Therefore, the two triangles are indeed congruent because triangle DEF can be mapped to triangle RPQ by a reflection across the x-axis followed by a translation 2 units left, which supports option 1 as the correct answer.