Question

1. find the measure of ∠abd and ∠dbc given m∠abc = 77°.

Explanation:

Step1: Set up an equation

Since $\angle ABC=\angle ABD+\angle DBC$, we have $(4x - 2)+(3x + 2)=77$.
Simplifying the left - hand side gives $4x-2 + 3x+2=7x$. So, $7x = 77$.

Step2: Solve for x

Dividing both sides of the equation $7x = 77$ by 7, we get $x=\frac{77}{7}=11$.

Step3: Find the measure of $\angle ABD$

Substitute $x = 11$ into the expression for $\angle ABD$. $\angle ABD=4x-2=4\times11 - 2=44 - 2=42^{\circ}$.

Step4: Find the measure of $\angle DBC$

Substitute $x = 11$ into the expression for $\angle DBC$. $\angle DBC=3x + 2=3\times11+2=33 + 2=35^{\circ}$.

Answer:

$\angle ABD = 42^{\circ}$, $\angle DBC=35^{\circ}$