Question

solving for exterior angle measures
what is the measure of ∠abd?
○ 27°
○ 54°
○ 114°
○ 124°
(diagram: triangle bcd with point b on a line ab, angle at c is 60°, angle at d is (2n)°, exterior angle at b (∠abd) is (4n + 6)°)

Explanation:

Step1: Use exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \(4n + 6=2n + 60\).

Step2: Solve for n

Subtract \(2n\) from both sides: \(4n-2n + 6=2n-2n + 60\), 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 \(\angle ABD\) (which is \(4n + 6\)): \(4\times27+6=108 + 6=114^{\circ}\).

Answer:

\(114^{\circ}\) (corresponding to the option \(114^{\circ}\))