To solve for $m \angle ABD$, we will use the given information and the relationship between the angles.

Step 1: Identify the Relationship Between Angles

From the diagram, we see that:

• $m \angle ABC + m \angle CBD = m \angle ABD$ because $\angle ABC$ and $\angle CBD$ are adjacent angles that together form $\angle ABD$.

Step 2: Write the Equation

Given the following expressions for the angles:

• $m \angle ABC = 3x$
• $m \angle CBD = x + 12$
• $m \angle ABD = 5x - 12$

The equation relating these angles is: $3x + (x + 12) = 5x - 12$

Step 3: Solve the Equation

Let's solve this equation for $x$.

$3x + x + 12 = 5x - 12$ $4x + 12 = 5x - 12$ Subtract $4x$ from both sides: $12 = x - 12$ Add 12 to both sides: $x = 24$

Step 4: Find $m \angle ABD$

Now that we have $x = 24$, substitute this value into the expression for $m \angle ABD$: $m \angle ABD = 5x - 12 = 5(24) - 12 = 120 - 12 = 108^\circ$

Final Answer:

The measure of $m \angle ABD$ is $108^\circ$.

Would you like further details or have any questions?

Here are 5 related questions:

1. How can you verify that the solution satisfies the original equation?
2. What would happen if the sum of the angles didn't add up correctly?
3. How do adjacent angles contribute to solving for unknowns in geometry?
4. How could this problem change if it involved supplementary angles?
5. What are the real-world applications of solving angle problems like this?

Tip: Always recheck your equation setup to ensure you're correctly representing the relationships between the angles.