Ray BE bisects ∠ABC. If m∠ABE = (3x + 8)° and m∠CBE = (7x − 8)°, what is m∠CBE?

Since ray \( BE \) bisects \(\angle ABC\), we know that:

\[
m\angle ABE = m\angle CBE
\]

Given that:

\[
m\angle ABE = (3x + 8)^\circ \quad \text{and} \quad m\angle CBE = (7x - 8)^\circ
\]

We set these two expressions equal to each other:

\[
3x + 8 = 7x - 8
\]

Now, solve for \( x \):

\[
8 + 8 = 7x - 3x
\]

\[
16 = 4x
\]

\[
x = 4
\]

Now substitute \( x = 4 \) back into the expression for \( m\angle CBE \):

\[
m\angle CBE = 7x - 8 = 7(4) - 8 = 28 - 8 = 20^\circ
\]

So, \( m\angle CBE = 20^\circ \).

Would you like more details on any part of the solution? Here are some related questions:

1. How do you prove that a ray bisects an angle?
2. What are the properties of an angle bisector?
3. How do you solve for \( x \) when two expressions are equal?
4. What is the relationship between adjacent angles when a line bisects an angle?
5. How do we verify the solution once we find the value of \( x \)?

**Tip:** When solving equations involving bisected angles, setting the two angles equal to each other is often the key step.

Learn how to solve an angle bisector problem where ray BE bisects ∠ABC. With m∠ABE = (3x + 8)° and m∠CBE = (7x − 8)°, find the value of x and m∠CBE. This problem involves algebraic equations and the angle bisector theorem, suitable for students in Grades 7-9.

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