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 ).

Explanation:

Step1: Apply angle addition postulate

By angle addition postulate, \(m\angle ABC=m\angle ABD + m\angle CBD\).
So, \(3x + 5=(2x + 10)+55\).

Step2: Solve for \(x\)

Simplify the equation:
\(3x+5=2x + 65\).
Subtract \(2x\) from both sides: \(3x-2x+5=2x-2x + 65\), which gives \(x+5=65\).
Subtract \(5\) from both sides: \(x=65 - 5=60\).

Step3: Find \(m\angle ABC\)

Substitute \(x = 60\) into \(m\angle ABC=3x + 5\).
\(m\angle ABC=3\times60+5=180 + 5=185\). But wait, this is wrong. Wait, no, actually, the correct angle - addition is \(m\angle ABD=m\angle ABC + m\angle CBD\) (from the figure, \(\angle ABC\) and \(\angle CBD\) form \(\angle ABD\)).
So, \(55=(3x + 5)+(2x + 10)\).
Combine like terms: \(55=5x+15\).
Subtract \(15\) from both sides: \(5x=55 - 15=40\).
Divide by \(5\): \(x = 8\).
Now find \(m\angle ABC\): substitute \(x = 8\) into \(m\angle ABC=3x + 5\).
\(m\angle ABC=3\times8+5=24 + 5=29\).

Answer:

\(x = 8\) and \(m\angle ABC=29^{\circ}\)