Question

find the measure of each marked angle.
(6x - 110)°
(x + 30)°
(6x - 50)°
the degree measure of the top angle of the triangle, with angle measure (6x - 110)°, is \\(\square\\)°.

Explanation:

Step1: Recall the exterior angle theorem

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. Here, the exterior angle is \((6x - 50)^{\circ}\), and the two non - adjacent interior angles are \((x + 30)^{\circ}\) and \((6x-110)^{\circ}\). So we have the equation:
\(6x - 50=(x + 30)+(6x-110)\)

Step2: Simplify the right - hand side of the equation

Simplify \((x + 30)+(6x-110)\):
\(x+30 + 6x-110=7x-80\)
So our equation becomes \(6x - 50=7x-80\)

Step3: Solve for \(x\)

Subtract \(6x\) from both sides:
\(6x-50 - 6x=7x - 80-6x\)
\(-50=x - 80\)
Add 80 to both sides:
\(-50 + 80=x-80 + 80\)
\(x = 30\)

Step4: Find the measure of the top angle

The top angle is \((6x-110)^{\circ}\). Substitute \(x = 30\) into the expression:
\(6\times30-110=180 - 110=70\)

Answer:

\(70\)