Question

use the diagram to the left to answer questions 9 and 10. 9. if (mangle abf=(6x + 26)^{circ}), (mangle ebf=(2x - 9)^{circ}), and (mangle abe=(11x - 31)^{circ}), find (mangle abf). 10. if (overrightarrow{bd}) bisects (angle cbe), (overrightarrow{bc}perpoverrightarrow{ba}), (mangle cbd=(3x + 25)^{circ}), and (mangle dbe=(7x - 19)^{circ}), find (mangle abd).

Explanation:

Step1: Use angle - addition postulate for Q9

$m\angle ABF + m\angle EBF=m\angle ABE$, so $(6x + 26)+(2x - 9)=11x - 31$.

Step2: Solve for x

$8x + 17=11x - 31$, $3x=48$, $x = 16$.

Step3: Find $m\angle ABF$ for Q9

$m\angle ABF=6x + 26=6\times16 + 26=96+26 = 122^{\circ}$.

Step4: Use angle - bisector property for Q10

Since $\overrightarrow{BD}$ bisects $\angle CBE$, $m\angle CBD=m\angle DBE$, so $3x + 25=7x - 19$.

Step5: Solve for x in Q10

$4x = 44$, $x = 11$.

Step6: Find $m\angle CBD$

$m\angle CBD=3x + 25=3\times11+25=33 + 25=58^{\circ}$.

Step7: Find $m\angle ABD$ for Q10

Since $\overrightarrow{BC}\perp\overrightarrow{BA}$, $m\angle ABC = 90^{\circ}$, $m\angle ABD=m\angle ABC+m\angle CBD=90^{\circ}+58^{\circ}=148^{\circ}$.

Answer:

1. $122^{\circ}$
2. $148^{\circ}$