Question

given: $m\angle abd = m\angle dbc$. find the value of x, and then find $m\angle abd$ and $m\angle abc$.

Explanation:

Step1: Set up equation

Since $m\angle ABD=m\angle DBC$, we have $2x + 6=5x-30$.

Step2: Solve for x

Subtract $2x$ from both sides: $6 = 5x-2x-30$, which simplifies to $6=3x - 30$. Then add 30 to both sides: $3x=6 + 30=36$. Divide both sides by 3, so $x = 12$.

Step3: Find $m\angle ABD$

Substitute $x = 12$ into the expression for $m\angle ABD$, $m\angle ABD=2x+6=2\times12 + 6=24 + 6=30$.

Step4: Find $m\angle ABC$

Since $m\angle ABC=m\angle ABD + m\angle DBC$ and $m\angle ABD=m\angle DBC = 30$, then $m\angle ABC=30+30 = 60$.

Answer:

$x = 12$
$m\angle ABD=30$
$m\angle ABC=60$