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.