Question

2 triangles are shown. Line M N is the line of reflection. Line segment X prime X has a midpoint at point A. Line segment Z prime Z has a midpoint at point B. Line segment Y prime Y has a midpoint at point C.
Which statements must be true about the reflection of ΔXYZ across Line M N? Select three options.

m∠X'Z'Y' = 90°
m∠MCY = 90°
XX' ≅ YY'
BZ' ≅ BZ
XY || X'Y'

Answer 1

Only two of the given statements are necessarily true for any reflection across line MN:

True
- m∠MCY = 90° — C is the midpoint of YY', so YY' is perpendicular to the mirror MN; CM lies on MN, so ∠MCY is a right angle.
- BZ' ≅ BZ — B is the midpoint of ZZ', so BZ = BZ' by definition.

Not necessarily true
- m∠X'Z'Y' = 90° — a reflection does not force the image triangle to be right unless the original triangle was right; this is not guaranteed.
- XX' ≅ YY' — XX' and YY' are twice the perpendicular distances of X and Y from MN; those distances need not be equal.
- XY || X'Y' — reflecting a segment generally produces a line symmetric to the original about MN, not necessarily parallel (only parallel in special cases).

So there are only two statements that must be true; the other three can be false depending on the positions of X, Y, Z.