Question

a c b d (8x)° (x + 27)°
____ = x
____ = m∠acd
____ = m∠dcb
____ = m∠acb

Explanation:

Step1: Use angle - addition postulate

Since $\angle ACB$ is a straight - angle, $m\angle ACD + m\angle DCB=180^{\circ}$. Given $m\angle ACD=(8x)^{\circ}$ and $m\angle DCB=(x + 27)^{\circ}$, we have the equation $8x+(x + 27)=180$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $8x+x+27 = 180$, which simplifies to $9x+27 = 180$.

Step3: Isolate the variable term

Subtract 27 from both sides of the equation: $9x+27-27=180 - 27$, resulting in $9x=153$.

Step4: Solve for x

Divide both sides of the equation by 9: $\frac{9x}{9}=\frac{153}{9}$, so $x = 17$.

Step5: Find $m\angle ACD$

Substitute $x = 17$ into the expression for $m\angle ACD$: $m\angle ACD=8x=8\times17 = 136^{\circ}$.

Step6: Find $m\angle DCB$

Substitute $x = 17$ into the expression for $m\angle DCB$: $m\angle DCB=x + 27=17+27 = 44^{\circ}$.

Step7: Find $m\angle ACB$

Since $\angle ACB$ is a straight - angle, $m\angle ACB = 180^{\circ}$.

Answer:

$x = 17$, $m\angle ACD=136^{\circ}$, $m\angle DCB = 44^{\circ}$, $m\angle ACB=180^{\circ}$