Question

in this figure, ( mangle a=(2x - 2)^{circ}, mangle c=(4x - 6)^{circ} ), and ( mangle dbc=(5x + 4)^{circ} ). what is ( mangle c )? 12 degrees 37 degrees 90 degrees 42 degrees

Explanation:

Step1: Apply 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. So, \(m\angle DBC=m\angle A + m\angle C\).
Substitute the given angle expressions: \((5x + 4)=(2x-2)+(4x - 6)\).

Step2: Solve the equation for \(x\)

Simplify the right - hand side: \((2x-2)+(4x - 6)=2x+4x-2 - 6=6x-8\).
The equation becomes \(5x + 4=6x-8\).
Subtract \(5x\) from both sides: \(4=x - 8\).
Add \(8\) to both sides: \(x=12\).

Step3: Find \(m\angle C\)

Substitute \(x = 12\) into the expression for \(m\angle C\).
\(m\angle C=(4x-6)^{\circ}\).
\(m\angle C=(4\times12 - 6)^{\circ}=(48-6)^{\circ}=42^{\circ}\).

Answer:

42 degrees