Question

1. if m∠abf=(6x + 26)°, m∠ebf=(2x - 9)°, and m∠abe=(11x - 31)°, find m∠abf.

Explanation:

Step1: Use angle - addition postulate

Since $\angle ABE=\angle ABF+\angle EBF$, we can set up the equation $(11x - 31)=(6x + 26)+(2x - 9)$.

Step2: Simplify the right - hand side of the equation

Combine like terms on the right - hand side: $(6x+26)+(2x - 9)=6x+2x+26 - 9=8x + 17$. So the equation becomes $11x-31 = 8x+17$.

Step3: Solve for $x$

Subtract $8x$ from both sides: $11x-8x-31=8x - 8x+17$, which simplifies to $3x-31 = 17$. Then add 31 to both sides: $3x-31 + 31=17 + 31$, so $3x=48$. Divide both sides by 3: $x = 16$.

Step4: Find $m\angle ABF$

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

Answer:

$122^{\circ}$