Question

part 3: angle addition & bisectors

1. if ( mangle abc = 3x + 5 ) and ( mangle cbd = 2x + 10 ), and ( mangle abd = 55^{circ} ), find ( x ) and ( mangle abc ).
2. ray ( bd ) bisects ( angle abc ). if ( mangle abd = 4x + 6 ) and ( mangle dbc = 6x - 4 ), find ( mangle abc ). **draw a picture.

Explanation:

Step1: Set up equation for angle bisector

Since ray \(BD\) bisects \(\angle ABC\), \(m\angle ABD=m\angle DBC\). So, \(4x + 6=6x-4\).

Step2: Solve for \(x\)

Subtract \(4x\) from both sides: \(6 = 2x-4\). Add \(4\) to both sides: \(10=2x\). Divide by \(2\): \(x = 5\).

Step3: Find \(m\angle ABD\)

Substitute \(x = 5\) into \(m\angle ABD=4x + 6\). \(m\angle ABD=4\times5+6=20 + 6=26^{\circ}\).

Step4: Calculate \(m\angle ABC\)

Since \(m\angle ABC=m\angle ABD+m\angle DBC\) and \(m\angle ABD=m\angle DBC\), \(m\angle ABC = 2\times m\angle ABD\). So, \(m\angle ABC=2\times26^{\circ}=52^{\circ}\).

Answer:

\(x = 5\) and \(m\angle ABC=52^{\circ}\)