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: Set up an equation for z

Since vertical angles are equal, we set $(5z - 16)=(3z + 32)$.
$5z-16 = 3z + 32$

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∠CEB

∠CEB and $(3z + 32)^{\circ}$ are supplementary. Substitute $z = 24$ into $(3z + 32)$: $3\times24+32=72 + 32=104$. So $m\angle CEB=180-(3z + 32)=180 - 104 = 76^{\circ}$.

Step4: Find m∠DEB

∠DEB and $(3z + 32)^{\circ}$ are vertical - angles. Substitute $z = 24$ into $(3z + 32)$: $3\times24+32=104^{\circ}$, so $m\angle DEB = 104^{\circ}$.

Step5: Set up an equation for x

Since ∠AEC and ∠DEB are vertical - angles, and ∠AEC=$(9x + 4)^{\circ}$ and ∠DEB = 104°, we set $9x+4=104$. Subtract 4 from both sides: $9x+4 - 4=104 - 4$, so $9x=100$. Divide both sides by 9: $x=\frac{100}{9}$.

Answer:

a. $z = 24$
b. $m\angle CEB=76^{\circ}$
c. $m\angle DEB = 104^{\circ}$
d. $x=\frac{100}{9}$