Question

in \\( \triangle a b c \\) shown below, side \\( \overline{a c} \\) is extended to point \\( d \\) with \\( m \angle d a b=(180-3 x)^{circ}, m \angle b=(6 x-40)^{circ} \\), and \\( m \angle c=(x+20)^{circ} \\).

what is \\( m \angle b a c \\)?

Explanation:

Step1: Use the exterior - angle theorem

The exterior - angle theorem states that \(m\angle DAB=m\angle B + m\angle C\).
Substitute the given angle measures: \((180 - 3x)=(6x - 40)+(x + 20)\).

Step2: Solve the equation for \(x\)

Simplify the right - hand side: \((180 - 3x)=7x-20\).
Add \(3x\) to both sides: \(180=10x - 20\).
Add \(20\) to both sides: \(200 = 10x\).
Divide both sides by \(10\): \(x = 20\).

Step3: Find \(m\angle BAC\)

Since \(m\angle DAB+m\angle BAC = 180^{\circ}\) (linear pair), and \(m\angle DAB=(180 - 3x)^{\circ}\).
Substitute \(x = 20\) into \(m\angle DAB\): \(m\angle DAB=(180-3\times20)^{\circ}=120^{\circ}\).
Then \(m\angle BAC=180 - m\angle DAB\).
\(m\angle BAC=180 - 120=60^{\circ}\).

Answer:

\(60^{\circ}\) (Option 3)