Question

in the figure, \\(\overrightarrow{ba}\\) and \\(\overrightarrow{bc}\\) are opposite rays. \\(\overrightarrow{bh}\\) bisects \\(\angle ebc\\) and \\(\overrightarrow{be}\\) bisects \\(\angle abf\\). 7. if \\(m\angle ebh=(6x + 12)^{circ}\\) and \\(m\angle hbc=(8x - 10)^{circ}\\), find \\(m\angle ebh\\). 8. if \\(m\angle abf=(7b - 24)^{circ}\\) and \\(m\angle abe = 2b^{circ}\\), find \\(m\angle ebf\\).

Explanation:

7.

Step1: Use angle - bisector property

Since $\overrightarrow{BH}$ bisects $\angle EBC$, then $m\angle EBH=m\angle HBC$. So we set up the equation $6x + 12=8x-10$.

Step2: Solve for $x$

Subtract $6x$ from both sides: $12 = 2x-10$. Then add 10 to both sides: $22=2x$. Divide both sides by 2, we get $x = 11$.

Step3: Find $m\angle EBH$

Substitute $x = 11$ into the expression for $m\angle EBH$. $m\angle EBH=(6x + 12)^{\circ}=(6\times11 + 12)^{\circ}=(66+12)^{\circ}=78^{\circ}$.

Step1: Use angle - bisector property

Since $\overrightarrow{BE}$ bisects $\angle ABF$, then $m\angle ABE=m\angle EBF$ and $m\angle ABF = 2m\angle ABE$. So we set up the equation $7b-24 = 2\times(2b)$.

Step2: Solve for $b$

Expand the right - hand side: $7b-24 = 4b$. Subtract $4b$ from both sides: $3b-24 = 0$. Then add 24 to both sides: $3b=24$. Divide both sides by 3, we get $b = 8$.

Step3: Find $m\angle EBF$

Since $m\angle EBF=m\angle ABE = 2b^{\circ}$, substitute $b = 8$ into it. $m\angle EBF=2\times8^{\circ}=16^{\circ}$.

Answer:

$78^{\circ}$

8.