BC bisects <ABD. If m<ABC=4x and M<CBD=x+60 find m<ABD

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.

Learn how to find angle ABD using the Angle Bisector Theorem in geometry. Given that BC bisects angle ABD with angle ABC = 4x and angle CBD = x + 60, discover the steps to calculate angle ABD effectively. Suitable for high school students studying geometry.

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