https://file.aiquickdraw.com/mathbot/file/1725911870945s5zq0ndy.jpg

To solve for \( m\angle ABD \) and \( m\angle CBD \), we have the following information from the image and the given equation:

1. \( m\angle ABC = 111^\circ \) (given).
2. \( m\angle ABD = (-10x + 58)^\circ \) (as labeled in the diagram).
3. \( m\angle CBD = (6x + 41)^\circ \) (as labeled in the diagram).

### Step 1: Apply the Angle Sum Property
We know that the sum of the angles \( m\angle ABD \) and \( m\angle CBD \) is equal to \( m\angle ABC \):
\[
m\angle ABD + m\angle CBD = m\angle ABC
\]
Substitute the given expressions for the angles:
\[
(-10x + 58) + (6x + 41) = 111
\]

### Step 2: Solve for \( x \)
Simplify the equation:
\[
-10x + 6x + 58 + 41 = 111
\]
\[
-4x + 99 = 111
\]
Now, solve for \( x \):
\[
-4x = 111 - 99
\]
\[
-4x = 12
\]
\[
x = -3
\]

### Step 3: Calculate \( m\angle ABD \) and \( m\angle CBD \)
Now that we have \( x = -3 \), substitute this value back into the expressions for \( m\angle ABD \) and \( m\angle CBD \).

- \( m\angle ABD = -10(-3) + 58 = 30 + 58 = 88^\circ \)
- \( m\angle CBD = 6(-3) + 41 = -18 + 41 = 23^\circ \)

### Final Answer:
\[
m\angle ABD = 88^\circ \quad \text{and} \quad m\angle CBD = 23^\circ
\]

Would you like further details or explanations on any part of this solution?

### Related Questions:
1. What are supplementary angles, and how do they apply to this problem?
2. How would the solution change if \( m\angle ABC \) were different?
3. Can you verify the angle calculations by adding \( m\angle ABD \) and \( m\angle CBD \)?
4. How can you check for calculation errors in solving for \( x \)?
5. What happens if you change the coefficients in the angle expressions?

### Tip:
Always double-check your algebraic steps when solving for variables in geometry to ensure accurate angle measurements.

Find m∠ABD and m∠CBD given m∠ABC = 111° and the angle expressions shown in the image.

Learn how to find the angles m∠ABD and m∠CBD given the angle m∠ABC = 111° using algebraic expressions and the angle sum property. This problem is ideal for students in Grades 8-10, focusing on key geometric and algebraic concepts.

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