Question

1. determine the value of x, m∠abd, and m∠abc.

Explanation:

Step1: Use angle - addition property

Since $\angle ABC=\angle ABD + \angle DBC$, we have the equation $(8x - 16)+(4x + 20)$.

Step2: Simplify the left - hand side of the equation

Combining like terms, we get $8x+4x-16 + 20=12x + 4$.
Assume $\angle ABC$ is a straight - angle ($180^{\circ}$) or we have some other relationship that gives us an equation to solve for $x$. If we assume $\angle ABC$ is a straight - angle:

Step3: Set up the equation

$12x+4 = 180$.

Step4: Solve for x

Subtract 4 from both sides: $12x=180 - 4=176$. Then $x=\frac{176}{12}=\frac{44}{3}$.

Step5: Find $\angle ABD$

Substitute $x = \frac{44}{3}$ into the expression for $\angle ABD$: $8x-16=8\times\frac{44}{3}-16=\frac{352}{3}-16=\frac{352 - 48}{3}=\frac{304}{3}\approx101.33^{\circ}$.

Step6: Find $\angle ABC$

Substitute $x=\frac{44}{3}$ into the expression for $\angle ABC$: $4x + 20=4\times\frac{44}{3}+20=\frac{176}{3}+20=\frac{176+60}{3}=\frac{236}{3}\approx78.67^{\circ}$.

Answer:

$x=\frac{44}{3}$, $m\angle ABD=\frac{304}{3}^{\circ}$, $m\angle ABC=\frac{236}{3}^{\circ}$