Question

given: $overline{mt}congoverline{lt}$, $overline{ot}$ is a bisector of $angle mtl$. prove: $overline{ot}perpoverline{lm}$ 1. $overline{mt}congoverline{lt}$, $overline{ot}$ is a bisector of $angle mtl$. 1. given 2. click here to insert 2. definition of an angle bisector 3. $overline{ot}congoverline{ot}$ 3. click here to insert 4. $\triangle mtocong\triangle lto$ 4. click here to insert

Explanation:

Step1: Apply angle - bisector definition

Since $\overline{OT}$ is a bisector of $\angle MTL$, by the definition of an angle bisector, $\angle MTO\cong\angle LTO$.

Step2: State reflexive property

$\overline{OT}\cong\overline{OT}$ by the reflexive property of congruence which states that any segment is congruent to itself.

Step3: Prove triangle congruence

We know that $\overline{MT}\cong\overline{LT}$ (given), $\angle MTO\cong\angle LTO$ (from step 1), and $\overline{OT}\cong\overline{OT}$ (from step 2). So, by the Side - Angle - Side (SAS) congruence criterion, $\triangle MTO\cong\triangle LTO$.

Answer:

1. $\angle MTO\cong\angle LTO$
2. Reflexive property of congruence
3. Side - Angle - Side (SAS) congruence criterion