Question

1. given the diagram below, find x and z.

a. solve for z.
b. find m∠ceb.
c. find m∠deb.
d. solve for x.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, $5z - 16=3z + 32$.

Step2: Solve the equation for z

Subtract $3z$ from both sides: $5z-3z - 16=3z-3z + 32$, which simplifies to $2z-16 = 32$. Then add 16 to both sides: $2z-16 + 16=32 + 16$, so $2z=48$. Divide both sides by 2: $z=\frac{48}{2}=24$.

Step3: Find $m\angle CEB$

Substitute $z = 24$ into the expression for $\angle CEB$. $m\angle CEB=3z + 32=3\times24+32=72 + 32=104^{\circ}$.

Step4: Find $m\angle DEB$

$\angle CEB$ and $\angle DEB$ are supplementary (linear - pair), so $m\angle DEB = 180 - m\angle CEB=180 - 104 = 76^{\circ}$.

Step5: Solve for x

Since $\angle AEC$ and $\angle DEB$ are vertical angles, $\angle AEC=\angle DEB$. So, $9x + 4=76$. Subtract 4 from both sides: $9x+4 - 4=76 - 4$, so $9x=72$. Divide both sides by 9: $x=\frac{72}{9}=8$.

Answer:

a. $z = 24$
b. $m\angle CEB=104^{\circ}$
c. $m\angle DEB=76^{\circ}$
d. $x = 8$