Question

find m∠abd, m∠cbd, and m∠abc.
a (6x + 14)°
d
(3x + 29)°
b c
m∠abd =
°
m∠cbd =
°
m∠abc =
°

Explanation:

Step1: Set up an equation

Assume $\angle ABC$ is a right - angle or there is some relationship that makes $\angle ABD+\angle CBD = 90^{\circ}$ (if not given otherwise, we assume $\angle ABC$ is a right - angle for the sake of solving with the given information). So, $(6x + 14)+(3x+29)=90$.

Step2: Combine like terms

$6x+3x+14 + 29=90$, which simplifies to $9x+43 = 90$.

Step3: Solve for $x$

Subtract 43 from both sides: $9x=90 - 43=47$, then $x=\frac{47}{9}$.

Step4: Find $m\angle ABD$

Substitute $x = \frac{47}{9}$ into the expression for $\angle ABD$: $m\angle ABD=6x + 14=6\times\frac{47}{9}+14=\frac{282}{9}+14=\frac{282 + 126}{9}=\frac{408}{9}=\frac{136}{3}\approx45.33^{\circ}$.

Step5: Find $m\angle CBD$

Substitute $x=\frac{47}{9}$ into the expression for $\angle CBD$: $m\angle CBD=3x + 29=3\times\frac{47}{9}+29=\frac{47}{3}+29=\frac{47+87}{3}=\frac{134}{3}\approx44.67^{\circ}$.

Step6: Find $m\angle ABC$

$m\angle ABC=m\angle ABD + m\angle CBD=\frac{136}{3}+\frac{134}{3}=\frac{136 + 134}{3}=\frac{270}{3}=90^{\circ}$

Answer:

$m\angle ABD=\frac{136}{3}^{\circ}$
$m\angle CBD=\frac{134}{3}^{\circ}$
$m\angle ABC = 90^{\circ}$