Question

which is a way to show △def is congruent to △def? a reflection across the y - axis and a translation 1 unit up and 1 unit left a rotation 180° about the origin and a translation 1 unit up a reflection across the line y = 1 and a translation 2 units up a rotation 90° and a translation 1 unit up and 1 unit left

Explanation:

Step1: Recall congruence - preserving transformations

Rigid - motions (reflections, rotations, translations) preserve congruence.

Step2: Analyze each option

• Option 1: A reflection across the y - axis changes the sign of the x - coordinates of the points of a figure. A translation 1 unit up and 1 unit left moves the figure. These are rigid - motions and preserve congruence.
• Option 2: A rotation 180° about the origin changes the signs of both x and y coordinates of the points of a figure. A translation 1 unit up is also a rigid - motion. These preserve congruence.
• Option 3: A reflection across the line y = 1 and a translation 2 units up are rigid - motions. The reflection flips the figure over the line y = 1 and the translation moves it up, and they preserve congruence.
• Option 4: A rotation 90° changes the orientation of the figure and a translation 1 unit up and 1 unit left. A 90 - degree rotation and a translation are rigid - motions and preserve congruence.

All of the given sequences of transformations are ways to show that two triangles are congruent since they are all composed of rigid - motions.

Answer:

All of the above (assuming the options are the four described transformation sequences)