Given that  
BA
  and  
BD
  are opposite rays, m∠CBA=12n+4, and m∠DBC=14n−6, find m∠DBC

Given that \( \overrightarrow{BA} \) and \( \overrightarrow{BD} \) are opposite rays, the angle \( \angle CBA \) and \( \angle DBC \) are supplementary. This is because the sum of the measures of the angles on a straight line is \(180^\circ\).

We are given:
\[
m\angle CBA = 12n + 4
\]
\[
m\angle DBC = 14n - 6
\]

Since \( \overrightarrow{BA} \) and \( \overrightarrow{BD} \) are opposite rays, \( m\angle CBA + m\angle DBC = 180^\circ \). Therefore, we can write the equation:
\[
(12n + 4) + (14n - 6) = 180
\]

Now, let's solve for \( n \):

1. Combine like terms:
\[
12n + 14n + 4 - 6 = 180
\]
\[
26n - 2 = 180
\]

2. Add 2 to both sides:
\[
26n = 182
\]

3. Divide by 26:
\[
n = \frac{182}{26} = 7
\]

Now, substitute \( n = 7 \) back into the expression for \( m\angle DBC \):
\[
m\angle DBC = 14n - 6 = 14(7) - 6 = 98 - 6 = 92^\circ
\]

So, \( m\angle DBC = 92^\circ \).

Would you like further details, or do you have any other questions?

Here are some related questions you might find interesting:
1. How would the problem change if \( m\angle CBA \) was \( 10n + 8 \) instead?
2. Can you explain why opposite rays form a straight angle?
3. What if \( \overrightarrow{BA} \) and \( \overrightarrow{BD} \) were not opposite rays, how would that affect the solution?
4. What are supplementary angles, and can you give other examples?
5. How do we find the measure of an angle given its supplementary or complementary angle?

**Tip:** When solving equations involving angles on a straight line, always check to ensure the sum equals \(180^\circ\).

Given that BA and BD are opposite rays, m∠CBA=12n+4, and m∠DBC=14n−6, find m∠DBC

Learn how to find the measure of angle DBC when given opposite rays BA and BD, and the measures of angles CBA and DBC in terms of n. This problem involves algebraic expressions, solving linear equations, and the concept of supplementary angles. Suitable for students in Grades 7-9.

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