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

We are given that \( BA \) and \( BD \) are opposite rays. This means that \( \angle CBA \) and \( \angle DBC \) form a straight line, making them supplementary. The sum of their angles will equal \( 180^\circ \).

The two angles are given as:
- \( m\angle CBA = 12n + 4 \)
- \( m\angle DBC = 14n - 6 \)

Since they are supplementary, we know:
\[
m\angle CBA + m\angle DBC = 180^\circ
\]

Substitute the expressions for the angles:
\[
(12n + 4) + (14n - 6) = 180
\]

Simplify the equation:
\[
12n + 14n + 4 - 6 = 180
\]
\[
26n - 2 = 180
\]

Solve for \( n \):
\[
26n = 182
\]
\[
n = \frac{182}{26} = 7
\]

Now that we have \( n = 7 \), we can find \( m\angle ABC \), which is formed by \( \angle CBA \) and \( \angle DBC \).

First, calculate \( m\angle CBA \) and \( m\angle DBC \):
\[
m\angle CBA = 12n + 4 = 12(7) + 4 = 84 + 4 = 88^\circ
\]
\[
m\angle DBC = 14n - 6 = 14(7) - 6 = 98 - 6 = 92^\circ
\]

Thus, \( m\angle ABC = m\angle CBA = 88^\circ \) (as they are part of the same angle).

### Do you want more details on any step or have any questions?
Here are 5 related questions to think about:
1. What are opposite rays, and why do they form supplementary angles?
2. How do you solve an equation with unknowns like \( n \) from angle measures?
3. What if \( BA \) and \( BD \) were not opposite rays, how would that change the problem?
4. How would the solution differ if the angles were complementary instead of supplementary?
5. What would happen if the value of \( n \) turned out to be a fraction?

**Tip:** When angles form a straight line, their sum is always \( 180^\circ \), a key concept in geometry!

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

Solve a geometry problem involving opposite rays BA and BD, with angle expressions m∠CBA=12n+4 and m∠DBC=14n−6. Learn how to find m∠ABC by using the supplementary angles theorem and solving a linear equation. Suitable for students in Grades 8-10.

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