Question

1. if m∠ebh=(6x + 12)° and m∠hbc=(8x - 10)°, find m∠ebh.
2. if m∠abf=(7b - 24)° and m∠abe = 2b°, find m∠ebf.
3. if m∠ebc=(31a - 2)° and m∠ebh=(4a + 45)°, find m∠hbc.

Explanation:

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Step1: Since $\angle EBH$ and $\angle HBC$ are adjacent angles and $\angle EBC=\angle EBH+\angle HBC$, assume $\angle EBC$ is a straight - angle ($180^{\circ}$). Then we set up the equation.

$(6x + 12)+(8x - 10)=180$

Step2: Combine like terms.

$6x+8x+12 - 10=180$
$14x + 2=180$

Step3: Isolate the variable term.

$14x=180 - 2$
$14x=178$

Step4: Solve for $x$.

$x=\frac{178}{14}=\frac{89}{7}$

Step5: Find $m\angle EBH$.

$m\angle EBH=(6x + 12)^{\circ}=6\times\frac{89}{7}+12=\frac{534}{7}+\frac{84}{7}=\frac{534 + 84}{7}=\frac{618}{7}\approx88.29^{\circ}$

Step1: Since $\angle ABF=\angle ABE+\angle EBF$, we can set up the equation to find $\angle EBF$.

$m\angle EBF=m\angle ABF - m\angle ABE$

Step2: Substitute the given expressions.

$m\angle EBF=(7b - 24)-2b$

Step3: Combine like terms.

$m\angle EBF=7b-2b - 24$
$m\angle EBF = 5b-24$

Step1: Since $\angle EBC=\angle EBH+\angle HBC$, we can find $\angle HBC$ by the formula $m\angle HBC=m\angle EBC - m\angle EBH$.

$m\angle HBC=(31a - 2)-(4a + 45)$

Step2: Distribute the negative sign.

$m\angle HBC=31a-2-4a - 45$

Step3: Combine like terms.

$m\angle HBC=31a-4a-2 - 45$
$m\angle HBC = 27a-47$

Answer:

$m\angle EBH=\frac{618}{7}^{\circ}$

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