Question

find m∠abd and m∠cbd given m∠abc = 111°. a (-10x + 58)° d (6x + 41)° b c m∠abd = □° and m∠cbd = □°.

Explanation:

Step1: Set up an equation

Since $\angle ABC=\angle ABD+\angle CBD$, we have $(- 10x + 58)+(6x + 41)=111$.

Step2: Simplify the left - hand side

Combine like terms: $(-10x+6x)+(58 + 41)=111$, which gives $-4x+99 = 111$.

Step3: Solve for x

Subtract 99 from both sides: $-4x=111 - 99$, so $-4x=12$. Then divide both sides by - 4, we get $x=-3$.

Step4: Find $m\angle ABD$

Substitute $x = - 3$ into the expression for $\angle ABD$: $m\angle ABD=-10x + 58=-10\times(-3)+58=30 + 58=88^{\circ}$.

Step5: Find $m\angle CBD$

Substitute $x=-3$ into the expression for $\angle CBD$: $m\angle CBD=6x + 41=6\times(-3)+41=-18 + 41 = 23^{\circ}$.

Answer:

$m\angle ABD = 88^{\circ}$ and $m\angle CBD=23^{\circ}$