Question

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

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{BD}$ 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: $25 = 4x-19$. Then add 19 to both sides: $44 = 4x$. Divide both sides by 4, we get $x = 11$.

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 $BC\perp BA$, then $\angle ABC = 90^{\circ}$. And $m\angle ABD=90^{\circ}-m\angle CBD$. Substitute $m\angle CBD = 58^{\circ}$, we get $m\angle ABD=90 - 58=32^{\circ}$.

Answer:

$32^{\circ}$