I was given the following question:
Polygon ABCD is a kite, with point E as the center and $\bar{AC}$ and $\bar {BD}$ as the diagonals. Which transformations can be used to proove that $\triangle ABC$ is congruent to $\triangle ADC$? Explain.
Then there were options given: A rotation 90 degrees around point c, a rotation 180 degrees around point e, a reflection over $\bar{AC}$, a tranlation from point b to point e.
Multiple answers can be chosen.
I don't understand how to do this. It seems to me that a reflection over $\bar{AC}$ will work because it just copies the kite onto itself. But I don't see how to prove it, and I can't figure out if any of the other ones work. Can anyone help me out?
It's given that $ABCD$ is a kite.
One of properties of a kite it's $AC\perp BD$ and $E$ is a middle-point of $BD$.
Thus, after reflection over $AC$ we have: $B$ goes to $D$, $D$ goes to $B$, $A$ goes to $A$ and $C$ goes to $C$.
Thus, $\Delta ABC$ goes to $\Delta ADC$ because after reflection a straight line goes to a straight line.
Id est, by the definition of the congruence $$\Delta ABC\cong\Delta ADC.$$