Question

applying into algebra
let \\( \overrightarrow { b d } \\) be the bisector of \\( \angle a b c \\).
find the requested values in the table
sketch on the diagram and show your thinking.

Explanation:

Step1: Use the angle - bisector property

Since \(BD\) is the bisector of \(\angle ABC\), then \(\angle ABD=\angle DBC\). So, \(4x = 3x + 6\).

Step2: Solve the equation for \(x\)

Subtract \(3x\) from both sides of the equation \(4x=3x + 6\).
\(4x-3x=3x + 6-3x\), which gives \(x = 6\).

Step3: Find \(\angle ABD\)

Substitute \(x = 6\) into the expression for \(\angle ABD\). Since \(\angle ABD=4x\), then \(\angle ABD=4\times6=24^{\circ}\).

Step4: Find \(\angle DBC\)

Substitute \(x = 6\) into the expression for \(\angle DBC\). Since \(\angle DBC=3x + 6\), then \(\angle DBC=3\times6+6=18 + 6=24^{\circ}\).

Step5: Find \(\angle ABC\)

Since \(\angle ABC=\angle ABD+\angle DBC\), and \(\angle ABD=\angle DBC = 24^{\circ}\), then \(\angle ABC=24^{\circ}+24^{\circ}=48^{\circ}\).

Answer:

Value	Check
\(\angle ABD\)	\(24^{\circ}\)
\(\angle DBC\)	\(24^{\circ}\)
\(\angle ABC\)	\(48^{\circ}\)