Question

given: $overline{mt}congoverline{lt}$, $overline{ot}$ is a bisector of $angle mtl$. prove: $overline{ot}perpoverline{lm}$ 5. corresponding parts of congruent tri - angles are congruent (c.p.c.t.c.) 6. definition of a linear pair 7. $angle mot$ and $angle lot$ are right angles. 8. $overline{ot}perpoverline{lm}$

Explanation:

Step1: Apply SAS congruence

We have two equal sides and the included equal angle in \(\triangle MOT\) and \(\triangle LOT\).

Step2: Get equal angles from CPCTC

Corresponding angles of congruent triangles are equal.

Step3: Use linear - pair sum

The sum of angles in a linear pair is \(180^{\circ}\).

Step4: Define perpendicular lines

Right - angled intersection implies perpendicularity.

Answer:

1. Step - 1: Prove \(\triangle MOT\cong\triangle LOT\)

• Since \(\overline{MT}\cong\overline{LT}\) (given), \(\angle MTO=\angle LTO\) (because \(\overline{OT}\) is the bisector of \(\angle MTL\) - given), and \(\overline{OT}=\overline{OT}\) (common side).
• By the Side - Angle - Side (SAS) congruence criterion, \(\triangle MOT\cong\triangle LOT\).

1. Step - 2: Use CPCTC

• Since \(\triangle MOT\cong\triangle LOT\), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC), \(\angle MOT=\angle LOT\).

1. Step - 3: Apply the linear - pair property

• \(\angle MOT\) and \(\angle LOT\) form a linear pair. By the definition of a linear pair, \(\angle MOT+\angle LOT = 180^{\circ}\).
• Since \(\angle MOT=\angle LOT\), let \(\angle MOT = x\) and \(\angle LOT=x\). Then \(x + x=180^{\circ}\), so \(2x = 180^{\circ}\), and \(x = 90^{\circ}\).

1. Step - 4: Conclude perpendicularity

• Since \(\angle MOT=\angle LOT = 90^{\circ}\), by the definition of perpendicular lines (if two lines intersect to form a right - angle, then they are perpendicular), \(\overline{OT}\perp\overline{LM}\).