Chapter 2.3, Problem 10CE

To determine

To fill :  In the blanks in the table given below by proving the angle bisector theorem.

Answer to Problem 10CE

Statement		Reasons
1.	$\overrightarrow{BX}$ is the bisector of $\angle ABC$	1.	Given in the statement.
2.	$\angle ABX\cong \angle XBC$ or $m\angle ABX\cong m\angle XBC$	2.	Angle bisector theorem
3.	$m\angle ABX+m\angle XBC=m\angle ABC$	3.	The angle addition postulate
4.	$m\angle ABX+m\angle ABX=m\angle ABC,$ or $2m\angle ABX=m\angle ABC$	4.	Substitution of $m\angle ABX=m\angle XBC$
5.	$m\angle ABX=\frac{1}{2}m\angle ABC$	5.	Dividing both sides by 2
6.	$m\angle XBC=\frac{1}{2}m\angle ABC$	6.	Substitution Property (Steps2 and5)

Explanation of Solution

Given information :  The information that is provided is that $\overrightarrow{BX}$ is the bisector of $\angle ABC$ .

Consider the figure given below, 

  

It is given that $\overrightarrow{BX}$ is the bisector of $\angle ABC$ . The Reason of Statement 1 is “Given in the statement.”

Angle bisector theorem states that angle bisector divides an angle into two parts or angles. As $\overrightarrow{BX}$ is the angle bisector, so it divides $\angle ABC$ into two parts or angles.

So, the Statement $\angle ABX\cong ?$ can be filled as $\angle ABX\cong \angle ABC$ . Also, the Statement $m\angle ABX\cong ?$ can be filled as $m\angle ABX\cong m\angle XBC$ ; the Reason of statement 2 is “Angle bisector theorem”.

Statement 3 is $m\angle ABX+m\angle XBC=m\angle ABC$ which is proved by the postulate “angle addition postulate”. This postulate is accepted without proof.

Substituting $m\angle ABX=m\angle XBC$ in step 3, you get

  $m\angle ABX+m\angle ABX=m\angle ABC,$ or $2m\angle ABX=m\angle ABC$

So, the reasoning for Statement 4 is “Substitution of $m\angle ABX=m\angle XBC$ in step 3.”

Dividing the equation in step 4 by 2, you get:

  $m\angle ABX=\frac{1}{2}m\angle ABC$

So, the reasoning for step 5 is “Divide both sides by 4”.

Supply “Substitution Property (Steps2 and5)” into the blank in the Reasoning for Statement 6. Substitute $\angle ABX\cong \angle XBC$ from step 2and $m\angle ABX=\frac{1}{2}m\angle ABC$ from step 5 which lead us to $m\angle XBC=\frac{1}{2}m\angle ABC$ .

The resultant table of two-column proof is an example of deductive reasoning.

Statement		Reasons
1.	$\overrightarrow{BX}$ is the bisector of $\angle ABC$	1.	Given in the statement.
2.	$\angle ABX\cong \angle XBC$ or $m\angle ABX\cong m\angle XBC$	2.	Angle bisector theorem
3.	$m\angle ABX+m\angle XBC=m\angle ABC$	3.	The angle addition postulate
4.	$m\angle ABX+m\angle ABX=m\angle ABC,$ or $2m\angle ABX=m\angle ABC$	4.	Substitution of $m\angle ABX=m\angle XBC$
5.	$m\angle ABX=\frac{1}{2}m\angle ABC$	5.	Dividing both sides by 2
6.	$m\angle XBC=\frac{1}{2}m\angle ABC$	6.	Substitution Property (Steps2 and5)