8. match the correct reasons to the proof:
given: bc is the bisector of $ \angle a b d $.
$ 6 n + 1 = 4 n + 19 $
$ 2 n = 18 $
$ n = 9 $
ray bc bisects $ \angle a b d $
$ m \angle a b c = m \angle c b d $
choices
definition of angle bisector
given
division property of equality
subtraction property of equality
substitution property of equality

Question

Accepted Answer

# Explanation:
## Step1: $6n + 1=4n + 19$
Since $BC$ is the bisector of $\angle ABD$, by the Definition of Angle Bisector, $m\angle ABC=m\angle CBD$. Substituting the given angle expressions $m\angle ABC=(6n + 1)^{\circ}$ and $m\angle CBD=(4n + 19)^{\circ}$, we get $6n + 1=4n + 19$ (Substitution Property of Equality).
## Step2: $2n=18$
Subtract $4n$ from both sides of $6n + 1=4n + 19$. Using the Subtraction Property of Equality: $(6n-4n)+1=(4n - 4n)+19$, so $2n+1 = 19$, then subtract $1$ from both sides: $2n+1-1=19 - 1$, resulting in $2n=18$.
## Step3: $n = 9$
Divide both sides of $2n=18$ by $2$. By the Division Property of Equality, $\frac{2n}{2}=\frac{18}{2}$, so $n = 9$.
## Step4: $ray\ BC$ bisects $\angle ABD$
This is given information.
## Step5: $m\angle ABC=m\angle CBD$
By the Definition of Angle Bisector (if a ray bisects an angle, then it divides the angle into two congruent angles).

# 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

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

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

Step2: \(2n=18\)

Step3: \(n = 9\)

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

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

Answer: