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

We are given that \( \overrightarrow{BA} \) and \( \overrightarrow{BD} \) are opposite rays, which means that together, they form a straight line. A straight line has an angle measure of \( 180^\circ \).

Since \( \angle CBA \) and \( \angle DBC \) are adjacent angles on a straight line, their sum must be \( 180^\circ \). Thus, we can write the equation:

\[
m\angle CBA + m\angle DBC = 180^\circ
\]

Substitute the given expressions for the angle measures:

\[
(12n + 4) + (14n - 6) = 180
\]

Simplify the left-hand side:

\[
12n + 4 + 14n - 6 = 180
\]

Combine like terms:

\[
26n - 2 = 180
\]

Add 2 to both sides:

\[
26n = 182
\]

Divide by 26:

\[
n = 7
\]

Now, substitute \( n = 7 \) into the expression for \( m\angle DBC \):

\[
m\angle DBC = 14n - 6 = 14(7) - 6 = 98 - 6 = 92^\circ
\]

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

Would you like further details or have any other questions? 

Here are five related questions for you:
1. What is the measure of \( m\angle CBA \) when \( n = 7 \)?
2. How would you prove that two rays are opposite?
3. Can opposite rays form angles greater than 180 degrees?
4. If one angle is known, how do you find the other adjacent angle on a straight line?
5. What would happen if the sum of the two angles was not 180 degrees?

**Tip**: Always check that the sum of angles on a straight line equals \( 180^\circ \), as it's a key property of linear pairs.

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

Discover how to solve for m∠DBC when BA and BD are opposite rays, with expressions m∠CBA = 12n + 4 and m∠DBC = 14n - 6. This problem involves using the Linear Pair Theorem and algebraic simplification, suitable for Grades 8-10.

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