We are asked to find three angle measurements: m∠ABC, m∠ABD, and m∠CBD. We have been told that BD bisects ∠ABC which means that it cuts the angle exactly in half. We have marked these relationships in the diagram below. Let's consider them.
Since BD bisects angle ∠ABC, angles ∠ABD and ∠DBC have the same measure. This means that m∠ABD=m∠DBC. Therefore, we can form an equation and substitute the given expressions for the measures to solve for x.
Having solved the equation, we can calculate the measures of individual angles by substituting x= - 8 into the given expressions.
m∠ABD:& - 4( -8)+33 =65^(∘)
m∠CBD:& 2 ( - 8)+81 =65^(∘)
Having found m∠ABD and m∠CBD, we can find m∠ABC. Since this angle is bisected by BD, its measure can be calculated by doubling either of the smaller angles.
m∠ABC: 65* 2 = 130^(∘)
