Question

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: Set up the equation

Let $m\angle ABD = x$. Then $m\angle ABC=3x - 1$. Since $\angle ABC=\angle ABD+\angle DBC$ and $\angle DBC = 47^{\circ}$, we have the equation $3x-1=x + 47$.

Step2: Solve for $x$

Subtract $x$ from both sides: $3x - x-1=x - x+47$, which simplifies to $2x-1 = 47$. Then add 1 to both sides: $2x-1 + 1=47 + 1$, so $2x=48$. Divide both sides by 2: $x=\frac{48}{2}=24$.

Step3: Find $m\angle ABC$

Substitute $x = 24$ into the expression for $m\angle ABC$. $m\angle ABC=3x-1=3\times24 - 1=72 - 1=71^{\circ}$.

Answer:

$m\angle ABD = 24^{\circ}$
$m\angle ABC = 71^{\circ}$