Question

Point Z is the incenter of ΔWXY.

Point Z is the incenter of triangle W X Y. Lines are drawn from point Z to the sides of the triangle to form line segments Z B, Z C, and Z A.
Which statements are true? Check all that apply.

m∠ZYX = m∠ZXY
ZX bisects ∠WXY
BZ = WZ
The circle inscribed in ΔWXY will have a center at point Z.
Point Z is equidistant from the sides of ΔWXY.

Answer 1

True:
- ZX bisects ∠WXY. (Z is the incenter, so it lies on the bisector of ∠X.)
- The circle inscribed in ΔWXY will have a center at point Z. (By definition the incenter is the center of the incircle.)
- Point Z is equidistant from the sides of ΔWXY. (All points on the angle bisectors that meet at the incenter are the same distance — the inradius — from each side.)

False:
- m∠ZYX = m∠ZXY. (Those are halves of ∠Y and ∠X respectively; they are equal only in special cases, not in general.)
- BZ = WZ. (BZ is an inradius segment (distance from Z to a side); WZ is the distance from Z to a vertex — not equal in general.)