Question

1. if ( mangle abc ) is one degree less than three times ( mangle abd ) and ( mangle dbc = 47^{circ} ), find each measure.

( mangle abd=)
( mangle abc=)

Explanation:

Step1: Establish the relationship between angles

Since \(m\angle ABC=m\angle ABD + m\angle DBC\), and \(m\angle ABC = 3m\angle ABD-1\), \(m\angle DBC = 47^{\circ}\).
So \(3m\angle ABD-1=m\angle ABD + 47\).

Step2: Solve the equation for \(m\angle ABD\)

Subtract \(m\angle ABD\) from both sides: \(3m\angle ABD - m\angle ABD-1=m\angle ABD - m\angle ABD+ 47\).
\(2m\angle ABD-1 = 47\).
Add \(1\) to both sides: \(2m\angle ABD-1 + 1=47 + 1\).
\(2m\angle ABD=48\).
Divide both sides by \(2\): \(m\angle ABD=\frac{48}{2}=24^{\circ}\).

Step3: Calculate \(m\angle ABC\)

Substitute \(m\angle ABD = 24^{\circ}\) into \(m\angle ABC=m\angle ABD + 47^{\circ}\).
\(m\angle ABC=24 + 47=71^{\circ}\).

Answer:

\(m\angle ABD = 24^{\circ}\), \(m\angle ABC = 71^{\circ}\)