Let's break down the problem:

1. BC bisects $\angle ABD$:

   • This means $\angle ABC = \angle CBD$.

2. Given values:

   • $\angle ABC = 4x$
   • $\angle CBD = x + 60$

Since BC bisects $\angle ABD$, the two angles $\angle ABC$ and $\angle CBD$ are equal. Therefore:

$4x = x + 60$

Step 1: Solve for $x$

Subtract $x$ from both sides:

$4x - x = 60 \implies 3x = 60 \implies x = 20$

Step 2: Find $\angle ABC$ and $\angle CBD$

Substitute $x = 20$ into the expressions for the angles:

$\angle ABC = 4x = 4(20) = 80^\circ$ $\angle CBD = x + 60 = 20 + 60 = 80^\circ$

Step 3: Calculate $\angle ABD$

Since $\angle ABD$ is made up of $\angle ABC$ and $\angle CBD$, we add these two angles together:

$\angle ABD = \angle ABC + \angle CBD = 80^\circ + 80^\circ = 160^\circ$

Final Answer:

$\angle ABD = 160^\circ$

Would you like more details or have any questions?

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Here are 5 related questions you might find interesting:

1. What would be the measure of $\angle ABD$ if $x$ were a different value?
2. How would the problem change if BC did not bisect $\angle ABD$?
3. Can you construct the angle bisector BC in a diagram with the given measures?
4. What is the sum of the interior angles in triangle ABC?
5. How does the concept of angle bisectors apply to other polygons?

Tip: When dealing with angle bisectors, remember that the bisector divides the angle into two equal parts, which is a key concept in many geometric problems.