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

From the reflection facts: A, B, C are midpoints of XX', ZZ', YY' and lie on the mirror line MN, and XX', YY', ZZ' are all perpendicular to MN.

So among the given choices:

- m∠MCY = 90° — True. C lies on MN, CY is perpendicular to MN, so angle MCY = 90°.
- BZ' ≅ BZ — True. B is the midpoint of ZZ', so BZ = BZ'.
- m∠X'Z'Y' = 90° — Not necessarily. Reflection preserves angle measures, but ∠X'Z'Y' equals ∠XZY; that angle need not be 90° unless the original was right.
- XX' ≅ YY' — Not necessarily. XX' = 2·(distance from X to MN) and YY' = 2·(distance from Y to MN); those distances can be different.
- XY || X'Y' — Not necessarily. A reflection does not in general produce a parallel corresponding side (only in special cases).

So only options 2 and 4 must be true. (If the problem really requires three selections, there is likely a mistake in the answer choices — the usual third necessary fact would be that corresponding sides are congruent, e.g. XY ≅ X'Y', which is not listed.)