Question

problem 22
given: $overline{mt}congoverline{lt}$, $overline{ot}$ is a bisector of $angle mtl$.
prove: $overline{ot}perpoverline{lm}$

1. $angle mot$ and $angle lot$ are a linear pair.
2. $angle mot$ and $angle lot$ are right angles.
3. $overline{ot}perpoverline{lm}$

Explanation:

Step1: Recall given information

Given $\overline{MT}\cong\overline{LT}$, $\overline{OT}$ is a bisector of $\angle MTL$.

Step2: Analyze step 6

In step 6, we know that $\angle MOT$ and $\angle LOT$ are a linear pair.

Step3: Determine the correct reason for step 7

Since $\angle MOT$ and $\angle LOT$ are a linear pair (step 6) and we want to show they are right - angles, the reason is that if a linear pair of angles are congruent, then they are right angles. This is because the sum of angles in a linear pair is $180^{\circ}$. If two congruent angles add up to $180^{\circ}$, then each angle is $\frac{180^{\circ}}{2}=90^{\circ}$, which is the definition of a right - angle.

Step4: Understand step 8

In step 8, since $\angle MOT$ and $\angle LOT$ are right angles, by the definition of perpendicular lines (two lines are perpendicular if they form right angles), we can conclude that $\overline{OT}\perp\overline{LM}$.

Answer:

We need to complete the proof. The correct reason for step 7 should be: "If a linear - pair of angles are congruent, then they are right angles."