Q. In given XYZ, XE is the bisector of angle YXZ. If YO perpendicular to XE and OM||XZ, prove that YM = MZ.

Q. In given XYZ, XE is the bisector of angle YXZ. If YO perpendicular to XE and OM||XZ, prove that YM = MZ.

Soln:
Constrution: Join O and A
1. In rs XYO and XAO
i. <YXO = <AXO (A)                       i. XE is bisector of <YXA
ii. XO = XO (S)                               ii. Common Side
iii. <YOX = <AOX                          iii. Both being right angle (i.e. 900)
iv. rXYO @ rXAO                       iv. From A.S.A. Axiom
2. YO = OA                                   2. Corresponding sides of @ rs.
3. YM = MZ                                  3. A straight line segment drawn through the mid-point of one side of a triangle and parallel to another side bisects third  side.