Question

1. ( overrightarrow{bd} ) bisects ( angle abc )( mangle abc=)____

Explanation:

Step1: Use angle - bisector property

Since \(\overrightarrow{BD}\) bisects \(\angle ABC\), then \(m\angle ABD=m\angle DBC\). So, \(3x + 6=4x\).

Step2: Solve for \(x\)

Subtract \(3x\) from both sides of the equation \(3x + 6=4x\). We get \(6=4x-3x\), so \(x = 6\).

Step3: Find \(m\angle ABC\)

Since \(m\angle ABC=m\angle ABD + m\angle DBC\) and \(m\angle ABD=m\angle DBC = 4x\) (or \(3x + 6\)), then \(m\angle ABC=(3x + 6)+4x=7x + 6\). Substitute \(x = 6\) into \(7x+6\). So, \(m\angle ABC=7\times6+6=42 + 6=48^{\circ}\).

Answer:

\(48^{\circ}\)