Question

1. if bd bisects ∠cbe, bc ⊥ ba, m∠cbd=(3x + 25)°, and m∠dbe=(7x - 19)°, find m∠abd.

Explanation:

Step1: Use angle - bisector property

Since $\overline{BD}$ bisects $\angle CBE$, then $\angle CBD=\angle DBE$. Given $\angle CBD=(3x + 25)^{\circ}$ and $\angle DBE=(7x-19)^{\circ}$, we set up the equation $3x + 25=7x-19$.

Step2: Solve the equation for $x$

Subtract $3x$ from both sides: $25 = 4x-19$. Then add 19 to both sides: $44 = 4x$. Divide both sides by 4, we get $x = 11$.

Step3: Find $\angle ABD$

Since $\angle ABD=\angle CBD$ (because $\overline{BD}$ bisects $\angle CBE$ and $\angle ABC = 90^{\circ}$ as $BC\perp BA$), substitute $x = 11$ into the expression for $\angle CBD$. $\angle CBD=3x + 25=3\times11+25=33 + 25=58^{\circ}$. So $\angle ABD = 58^{\circ}$.

Answer:

$58^{\circ}$