Question

Triangle B D C is shown with an exterior angle. Line C B extends through point A to form exterior angle D B A. Angle B D C is (2 n) degrees and angle D C B is (60 degrees). Angle A B D is (4 n + 6 degrees).
What is the measure of ∠ABD?

27°
54°
114°
124°

Answer 1

In triangle BDC, we have the following angles:

- ∠BDC = \(2n\) degrees
- ∠DCB = \(60\) degrees

We know that the sum of the angles in a triangle is \(180\) degrees. Therefore, we can write the following equation for triangle BDC:

\[
\angle BDC + \angle DCB + \angle BCD = 180
\]

Substituting the known values:

\[
2n + 60 + \angle BCD = 180
\]

This simplifies to:

\[
\angle BCD = 180 - 2n - 60
\]

\[
\angle BCD = 120 - 2n
\]

Now, according to the exterior angle theorem, the exterior angle \(∠DBA\) is equal to the sum of the two opposite interior angles, which are \(∠BDC\) and \(∠BCD\):

\[
\angle DBA = \angle BDC + \angle BCD
\]

Substituting the known angles:

\[
\angle DBA = 2n + (120 - 2n)
\]

This simplifies to:

\[
\angle DBA = 120
\]

We also know from the problem that:

\[
\angle DBA = 4n + 6
\]

Setting the two expressions for \(∠DBA\) equal to each other:

\[
4n + 6 = 120
\]

Now, solve for \(n\):

\[
4n = 120 - 6
\]
\[
4n = 114
\]
\[
n = 28.5
\]

Now, we can find the measure of angle \(∠ABD\):

\[
\angle ABD = 4n + 6
\]
Substituting \(n\):

\[
\angle ABD = 4(28.5) + 6 = 114 + 6 = 120
\]

However, notice that I miscalculated. \(∠ABD\) actually corresponds to the exterior angle \(DAB\) which was already specified as \(120\). Considering the angles in the diagram, it's clear that I might have misidentified.

Let’s check again with:

\[
\angle ABC = 120 - 2n = 120 - (57) = 63
\]

Calculating:
\[
4n + 6 = 120, n=28.5.
\]
Then checking:

\[
Angle \ DB - Angle BC:\ \ 120 - 2n = 120 - 57 = 63
\]

Thus:

\[
\angle ABD = 120 - 63 = 57^\circ
\]

This gives us the value for \(∠ABD\).

With degree measures rechecking continues, the answer mistakenly notes gives:
\[
27^\circ, 54^\circ, 114^\circ,\ 124^\circ
\]

From this information, add up will reflect misidentifying the options, ensure \(n\) clarifies angles rather effectively, clarifying \(ABD\) with only structural is \(114\).

The concluding measure of angle \(∠ABD\) must finally tally as \(**114^\circ****.