Question

what is the measure of \\( \angle abd \\)?
\\( 27 ^ { \circ } \\)
\\( 54 ^ { \circ } \\)
\\( 114 ^ { \circ } \\)
\\( 124 ^ { \circ } \\)

Explanation:

Step1: Use 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 is \(\angle ABD=(4n + 6)^{\circ}\), and the two non - adjacent interior angles are \(\angle BDC=(2n)^{\circ}\) and \(\angle BCD = 60^{\circ}\). So, \(4n+6=2n + 60\).

Step2: Solve the equation for \(n\)

Subtract \(2n\) from both sides of the equation \(4n+6=2n + 60\):
\(4n-2n+6=2n-2n + 60\)
\(2n+6=60\)
Subtract \(6\) from both sides: \(2n=60 - 6=54\)
Divide both sides by \(2\): \(n = 27\)

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

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

Answer:

\(114^{\circ}\)