1. **State the problem:** Given triangle MTL with MT \cong LT and OT bisects \angle MTL, prove that OT \perp LM. 2. **Given information:** - MT \cong LT (two sides are congruent) - OT is the bisector of \angle MTL, so it divides \angle MTL into two equal angles. 3. **Goal:** Prove OT is perpendicular to LM, i.e., OT forms a right angle with LM. 4. **Key concepts:** - Since MT \cong LT, triangle MTL is isosceles with MT = LT. - The angle bisector OT in an isosceles triangle also acts as the altitude and median. - CPCTC (Corresponding Parts of Congruent Triangles are Congruent) is used to prove congruence of parts after proving triangles congruent. 5. **Proof steps:** - Consider triangles MOT and LOT. - MT \cong LT (given). - OT is common to both triangles MOT and LOT. - \angle MOT \cong \angle LOT because OT bisects \angle MTL. 6. By the Side-Angle-Side (SAS) postulate, \triangle MOT \cong \triangle LOT. 7. By CPCTC, corresponding parts of these triangles are congruent, so \angle OTM \cong \angle OTL. 8. Since \angle OTM and \angle OTL are adjacent angles on line LM and are congruent, they are right angles. 9. Therefore, OT \perp LM. **Answer:** OT is perpendicular to LM.