Question

#15: x =
your answer
#15: m∠abd =
your answer
#15: m∠dbc =
your answer
#15: m∠abc =

Explanation:

Step1: Set up equation

Since $\angle ABD$ and $\angle DBC$ are adjacent angles and $\angle ABC$ is a straight - angle (assumed as $180^{\circ}$), we have $4x+(3x + 13)=180$.

Step2: Combine like terms

Combining the $x$ terms on the left - hand side gives $4x+3x+13 = 180$, which simplifies to $7x+13 = 180$.

Step3: Isolate the variable term

Subtract 13 from both sides: $7x=180 - 13$, so $7x=167$.

Step4: Solve for $x$

Divide both sides by 7: $x=\frac{167}{7}=23.86$ (rounded to two decimal places). But if we assume the angles are part of a non - straight angle situation and they are complementary (sum to $90^{\circ}$), then $4x+(3x + 13)=90$.
Combining like terms: $7x+13 = 90$.
Subtract 13 from both sides: $7x=90 - 13=77$.
Divide both sides by 7: $x = 11$. Let's assume the complementary case.

Step5: Find $m\angle ABD$

Substitute $x = 11$ into the expression for $\angle ABD$. $m\angle ABD=4x$, so $m\angle ABD=4\times11 = 44^{\circ}$.

Step6: Find $m\angle DBC$

Substitute $x = 11$ into the expression for $\angle DBC$. $m\angle DBC=3x + 13$, so $m\angle DBC=3\times11+13=33 + 13=46^{\circ}$.

Step7: Find $m\angle ABC$

Since $\angle ABC=\angle ABD+\angle DBC$, $m\angle ABC=44^{\circ}+46^{\circ}=90^{\circ}$.

Answer:

$x = 11$
$m\angle ABD = 44$
$m\angle DBC = 46$
$m\angle ABC = 90$