Question

solving for exterior angle measures
what is the measure of ∠abd?
27°
54°
114°
124°

Explanation:

Step1: Apply the exterior - angle theorem

The exterior - angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles.
For \(\triangle BCD\), the exterior angle \(\angle ABD=(2n)^{\circ}+60^{\circ}\), and we also know that \(\angle ABD=(4n + 6)^{\circ}\). So, \(4n+6=2n + 60\).

Step2: Solve the equation for \(n\)

Subtract \(2n\) from both sides of the equation \(4n+6=2n + 60\):
\(4n-2n+6=2n-2n + 60\), which simplifies to \(2n+6=60\).
Then subtract \(6\) from both sides: \(2n+6 - 6=60 - 6\), so \(2n=54\).
Divide both sides by \(2\): \(n = 27\).

Step3: Find the measure of \(\angle ABD\)

Substitute \(n = 27\) into the expression for \(\angle ABD=(4n + 6)^{\circ}\).
\(\angle ABD=4\times27+6\).
First, calculate \(4\times27 = 108\), then \(108+6=114\).

Answer:

\(114^{\circ}\)