Question

what is the measure of \\(\angle abd\\)?
\\(\bigcirc\\) \\(27^\circ\\)
\\(\bigcirc\\) \\(54^\circ\\)
\\(\bigcirc\\) \\(114^\circ\\)
\\(\bigcirc\\) \\(124^\circ\\)
(image shows a diagram with points a, b, c, d. angle at a is \\((4n + 6)^\circ\\), angle at d is \\((2n)^\circ\\), angle at c is \\(60^\circ\\), with a line from a through b to c, and lines from b to d and d to c.)

Explanation:

Step1: Identify triangle angle sum

$\angle DBC = 180^\circ - (2n)^\circ - 60^\circ = 120^\circ - 2n$
$\angle ABD + \angle DBC = 180^\circ$ (straight line at B)
$\angle ABD = (4n + 6)^\circ$ (given), so:
$(4n + 6) + (120 - 2n) = 180$

Step2: Solve for n

$2n + 126 = 180$
$2n = 54$
$n = 27$

Step3: Calculate ∠ABD

$\angle ABD = 4(27) + 6 = 108 + 6 = 114^\circ$

Answer:

C. 114°