Question

1. if $overline{bd}$ bisects $angle cbe$, $overline{bc}perpoverline{ba}$, $mangle cbd=(3x + 25)^{circ}$, and $mangle dbe=(7x - 19)^{circ}$, find $mangle abd$.

Explanation:

Step1: Use angle - bisector property

Since $\overline{BD}$ bisects $\angle CBE$, then $m\angle CBD=m\angle DBE$. So we set up the equation $3x + 25=7x-19$.

Step2: Solve the equation for $x$

Subtract $3x$ from both sides: $25 = 7x-3x - 19$, which simplifies to $25=4x - 19$. Then add 19 to both sides: $25 + 19=4x$, so $44 = 4x$. Divide both sides by 4: $x = 11$.

Step3: Find $m\angle CBD$

Substitute $x = 11$ into the expression for $m\angle CBD$: $m\angle CBD=3x + 25=3\times11+25=33 + 25=58^{\circ}$.

Step4: Find $m\angle ABD$

Since $\overline{BC}\perp\overline{BA}$, $m\angle ABC = 90^{\circ}$. Then $m\angle ABD=m\angle ABC - m\angle CBD$. So $m\angle ABD=90^{\circ}-58^{\circ}=32^{\circ}$.

Answer:

$32^{\circ}$