50 POINTS Preston says the sequence of transformations that maps triangle ABC onto triangle DEF is: rotate it 180 degrees about the origin, and then translate it up 7 units.

Chanel says the sequence of transformations that maps triangle ABC onto triangle DEF is: reflect it across the x-axis, then reflect it across the y-axis, and then translate it up 6 units and left 1 unit.

Part E
Now look at Chanel’s sequence. How does a reflection across the x-axis change the coordinates of a shape?

Part F
Complete the table to find the rule for the reflection, the coordinates of triangle ABC, and the coordinates of the reflected image, triangle A′B′C′.
Original Coordinates New Coordinates
(x, y) ( , )
A( , ) A′( , )
B( , ) B′( , )
C( , ) C′( , )

Part H
Complete the table to find the rule for the reflection, the coordinates of triangle A′B′C′, and the coordinates of the reflected image, triangle A″B″C″.
Original Coordinates New Coordinates
(x, y) ( , )
A'( , ) A"( , )
B'( , ) B"( , )
C'( , ) C"( , )

Part I
How does a translations 6 units up and 1 unit left change the coordinates of a shape?

Part J
Complete the table to find the rule for the translation, the coordinates of triangle A″B″C″, and the coordinates of the translated image, triangle A‴B‴C‴.

Original Coordinates New Coordinates
(x, y) ( , )
A'( , ) A"'( , )
B'( , ) B"'( , )
C'( , ) C"'( , )

Part K
Whose sequence proves triangle ABC is congruent to triangle DEF? Why?