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). ((6x + 26)+(2x - 9)=11x - 31) (8x+17 = 11x - 31) (17 = 3x - 31) (48 = 3x) (x = 16) (6(16)+26) (96 + 26) (mangle abf = 122) 10. 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 - addition postulate for question 9

Since $\angle ABE=\angle ABF+\angle EBF$, we have the equation $(6x + 26)+(2x - 9)=11x-31$.
Combining like - terms on the left - hand side gives $8x + 17=11x-31$.

Step2: Solve the equation for $x$

Subtract $8x$ from both sides: $17 = 3x-31$.
Add 31 to both sides: $48=3x$.
Divide both sides by 3: $x = 16$.

Step3: Find $m\angle ABF$

Substitute $x = 16$ into the expression for $m\angle ABF$: $m\angle ABF=6x + 26=6\times16+26=96 + 26=122^{\circ}$.

Step4: Use angle - bisector property for question 10

Since $\overline{BD}$ bisects $\angle CBE$, then $m\angle CBD=m\angle DBE$.
Set up the equation $3x + 25=7x-19$.
Subtract $3x$ from both sides: $25 = 4x-19$.
Add 19 to both sides: $44 = 4x$.
Divide both sides by 4: $x = 11$.

Step5: Find $m\angle CBD$

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

Step6: Find $m\angle ABC$

Since $\overline{BC}\perp\overline{BA}$, $m\angle ABC = 90^{\circ}$.

Step7: Find $m\angle ABD$

$m\angle ABD=m\angle ABC+m\angle CBD$.
So $m\angle ABD=90^{\circ}+58^{\circ}=148^{\circ}$.

Answer:

1. $m\angle ABF = 122^{\circ}$
2. $m\angle ABD = 148^{\circ}$