Question

1. $overline{bd}$ bisects $angle abc$, $mangle abd=(4x - 2)^{circ}$, $mangle dbc=(3x + 18)^{circ}$. find $mangle abc$. explain!

Explanation:

Step1: Set equal angles

Since $\overline{BD}$ bisects $\angle ABC$, then $m\angle ABD=m\angle DBC$. So we set up the equation $4x - 2=3x + 18$.
$4x-2=3x + 18$

Step2: Solve for x

Subtract $3x$ from both sides of the equation: $4x-3x-2=3x-3x + 18$, which simplifies to $x-2=18$. Then add 2 to both sides: $x=18 + 2$, so $x = 20$.
$x=20$

Step3: Find $m\angle ABD$ or $m\angle DBC$

Substitute $x = 20$ into the expression for $m\angle ABD$ (we could also use the expression for $m\angle DBC$). $m\angle ABD=4x-2=4\times20-2=80 - 2=78^{\circ}$.
$m\angle ABD = 78^{\circ}$

Step4: Calculate $m\angle ABC$

Since $m\angle ABC=m\angle ABD + m\angle DBC$ and $m\angle ABD=m\angle DBC$, then $m\angle ABC=2\times m\angle ABD$. So $m\angle ABC=2\times78^{\circ}=156^{\circ}$.
$m\angle ABC=156^{\circ}$

Answer:

$156^{\circ}$