Question

1. match the correct reasons to the proof:

given: bc is the bisector of ∠abd.
6n + 1 = 4n + 19
2n = 18
n = 9
ray bc bisects ∠abd
m∠abc = m∠cbd
choices
definition of angle bisector
given
division property of equality
subtraction property of equality
substitution property of equality

Explanation:

Step1: Match \(6n + 1=4n + 19\)

Since \(BC\) is the bisector of \(\angle ABD\), by the Definition of Angle Bisector, \(\angle ABC=\angle CBD\). So \(6n + 1 = 4n+19\) (Substitution Property of Equality).

Step2: Match \(2n = 18\)

Subtract \(4n\) from both sides of \(6n + 1=4n + 19\). Using the Subtraction Property of Equality: \((6n-4n)+1=(4n - 4n)+19\), which simplifies to \(2n+1 = 19\), then subtract \(1\) from both sides \(2n=18\).

Step3: Match \(n = 9\)

Divide both sides of \(2n = 18\) by \(2\). By the Division Property of Equality, \(n=\frac{18}{2}=9\).

Step4: Match \(ray\ BC\) bisects \(\angle ABD\)

This is given information.

Step5: Match \(m\angle ABC=m\angle CBD\)

This is from the Definition of Angle Bisector.

Answer:

\(6n + 1=4n + 19\): Substitution Property of Equality

\(2n = 18\): Subtraction Property of Equality

\(n = 9\): Division Property of Equality

\(ray\ BC\) bisects \(\angle ABD\): Given

\(m\angle ABC=m\angle CBD\): Definition of Angle Bisector