Question

given: ( overline{mn} ) is an angle bisector of ( angle jmk ) prove ( mangle jmn=\frac{1}{2}mangle jmk ) 1. ( overline{mn} ) is an angle bisector of ( angle jmk ) 1 given 2. ( angle jmncongangle nmk ) 2 3. ( mangle jmn = mangle nmk ) 3 definition of congruent angles 4. ( mangle jmn + mangle nmk = mangle jmk ) 4 angle addition postulate 5. ( mangle jmn + mangle jmn = mangle jmk ) 5 6. ( 2mangle jmn = mangle jmk ) 6 7. ( mangle jmn=\frac{1}{2}mangle jmk ) 7 division property of equality

Explanation:

Step1: Reason for Statement 2

By the definition of an angle bisector, if a ray bisects an angle, it divides the angle into two congruent angles. So, the reason for \( \angle JMN\cong\angle NMK\) (Statement 2) is "Definition of Angle Bisector".

Step2: Reason for Statement 5

Since \(m\angle JMN = m\angle NMK\) (Statement 3), we can substitute \(m\angle NMK\) with \(m\angle JMN\) in the equation \(m\angle JMN + m\angle NMK=m\angle JMK\) (Statement 4). So, the reason for \(m\angle JMN + m\angle JMN=m\angle JMK\) (Statement 5) is "Substitution".

Step3: Reason for Statement 6

Combining like - terms in \(m\angle JMN + m\angle JMN=m\angle JMK\), we use the simplification rule for adding like terms. So, the reason for \(2m\angle JMN=m\angle JMK\) (Statement 6) is "Simplify".

Answer:

1. Definition of Angle Bisector
2. Substitution
3. Simplify