Chapter 4.3, Problem 10WE

To determine

To prove: $\overrightarrow{LO}$ bisects $\angle MLN$

Explanation of Solution

Given information: $\overline{LM}\cong \overline{LN},\overline{KM}\cong \overline{KN}$ and $\overrightarrow{KO}$ bisects $\angle MKN$

Proof:

  

Consider the following definitions and properties:

Reflexive property of congruence for line segments states that line segments are congruent to each other.

SSS congruence rule states that two triangles are said to be congruent if all sides of one triangle are congruent to the corresponding sides of another triangle.

An angle bisector of an angle divides it into two congruent angles.

Use the above definitions and properties in the following two-column proof.

Statements	Reasons
1. $\overline{LM}\cong \overline{LN}$	1. Given
2. $\overline{LK}\cong \overline{LK}$	2. Reflexive property of congruence
3 $\overline{KM}\cong \overline{KN}$	3. Given
4 $\Delta LKM\cong \Delta LKN$	4. SSS congruence rule
5 $\angle KLM\cong \angle KLN$	5. CPCT
6 $\overrightarrow{LO}$ bisects $\angle MLN$	6. Definition of angle bisector

Here, the given segment ‘ $\overrightarrow{KO}$ bisects $\angle MKN$ ’ is not required.