1. **State the problem:** Determine which sequences of transformations prove congruency by mapping polygon I onto polygon II. 2. **Analyze Shape Locations:** Polygon I is in quadrant I around points like (2,8), (10,10). Polygon II is in quadrant III around points like (-12,-4), (-8,-6). 3. **Key idea:** To map I onto II, transformations must move from quadrant I to quadrant III (diagonally opposite). 4. **Transformation types:** - Reflection across y-axis: flips x-coordinates sign. - Rotation 90° CW: rotates points 90 degrees clockwise around origin. - Rotation 90° CCW: rotates 90 degrees counterclockwise. - Rotation 180°: rotates points 180 degrees. - Translation moves points in specific directions. 5. **Check sequences:** - a) Reflection across y-axis, then 90° CW rotation, then translate left 18 units. -- Reflection across y-axis changes quadrant I points to quadrant II. -- 90° CW rotation moves points from quadrant II to quadrant III. -- Translation left 18 units shifts all points to the left. -- This sequence can map shape I onto II, proving congruency. - b) 90° CW rotation, then translate left 18 units. -- 90° CW rotation moves quadrant I points to quadrant IV. -- Translation left 18 units moves points left but does not move to quadrant III. -- This does not map shape I onto shape II properly. - c) 90° CCW rotation, reflection across y-axis, then translate left 18 units. -- 90° CCW rotation moves quadrant I points to quadrant II. -- Reflection across y-axis reflects quadrant II to quadrant I. -- Overall, shape remains near quadrant I; translation left cannot move shape to III. -- Does not prove congruency. - d) 90° CW rotation, reflection across y-axis, then translate right 2 units. -- 90° CW rotation moves quadrant I to quadrant IV. -- Reflection across y-axis moves quadrant IV to quadrant III. -- Translation right 2 units shifts shape slightly, still in quadrant III. -- This sequence possibly maps I onto II, proving congruency. - e) 180° rotation, reflection across y-axis, then 90° CW rotation. -- 180° rotation moves quadrant I points to quadrant III. -- Reflection across y-axis moves quadrant III to quadrant IV. -- 90° CW rotation moves quadrant IV to quadrant I. -- Result ends in quadrant I, so does not map to II. - f) Reflection across y-axis, 90° CCW rotation, translate right 2 units. -- Reflection across y-axis moves I to II. -- 90° CCW rotation moves II to I. -- Translation right 2 only small shift. -- Does not map I onto II. 6. **Final classification:** Sequences that prove congruency by mapping I onto II are (a) and (d). **Answer:** The sequences (a) and (d) prove congruency by successfully mapping polygon I onto polygon II.