Question

a e b
____ = x
____ = m∠aec
____ = m∠bec
d
(6x + 2)°
(2x + 8)°
c

Explanation:

Step1: Use angle - sum property

Since $\angle AED = 90^{\circ}$ and $\angle AEB$ is a straight - angle ($180^{\circ}$), and $\angle AEC+\angle BEC = 180^{\circ}$, also $\angle AEC=\angle AED+\angle DEC$. Given $\angle DEC=(6x + 2)^{\circ}$ and $\angle BEC=(2x + 8)^{\circ}$, and $\angle AED = 90^{\circ}$, we know that $\angle AEC=90+(6x + 2)=(6x + 92)^{\circ}$. Then, $(6x + 92)+(2x + 8)=180$.

Step2: Solve the equation for x

Combine like - terms in the equation $(6x + 92)+(2x + 8)=180$. We get $6x+2x+92 + 8=180$, which simplifies to $8x+100 = 180$. Subtract 100 from both sides: $8x=180 - 100=80$. Divide both sides by 8: $x=\frac{80}{8}=10$.

Step3: Find $\angle AEC$

Substitute $x = 10$ into the expression for $\angle AEC$. $\angle AEC=6x+92$. So, $\angle AEC=6\times10 + 92=152^{\circ}$.

Step4: Find $\angle BEC$

Substitute $x = 10$ into the expression for $\angle BEC$. $\angle BEC=2x + 8$. So, $\angle BEC=2\times10+8=28^{\circ}$.

Answer:

$x = 10$, $m\angle AEC=152^{\circ}$, $m\angle BEC=28^{\circ}$