Question

what series of transformations map △abc onto △def to prove that △abc≅△def?
a reflection across x - axis then a positive rotation of 180° about the origin
a clockwise rotation of 180° about the origin then a translation of 1 unit right and 3 units up
a reflection across y - axis then translation of 1 unit right and 1 unit down
a reflection across y - axis then translation of 1 unit right and 2 units up

Explanation:

Step1: Analyze reflection across y - axis

A reflection across the y - axis changes the sign of the x - coordinate of each point. For example, if a point $(x,y)$ is reflected across the y - axis, it becomes $(-x,y)$.

Step2: Analyze translations

A translation of 1 unit right means adding 1 to the x - coordinate of each point, and a translation of 2 units up means adding 2 to the y - coordinate of each point.

Step3: Check congruence

By performing a reflection across the y - axis and then a translation of 1 unit right and 2 units up on $\triangle ABC$, we can map it onto $\triangle DEF$ to prove their congruence.

Answer:

a reflection across y - axis then translation of 1 unit right and 2 units up