Question

given the figure below, find the values of x and z.
(15x - 52)°
(10x - 43)°
z°
x=
z=

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, $15x - 52=10x - 43$.

Step2: Solve for $x$

Subtract $10x$ from both sides: $15x-10x - 52=10x-10x - 43$, which simplifies to $5x-52=-43$. Then add 52 to both sides: $5x-52 + 52=-43 + 52$, so $5x = 9$, and $x=\frac{9}{5}=1.8$.

Step3: Find the value of one of the angles

Substitute $x = 1.8$ into $15x-52$: $15\times1.8-52=27 - 52=-25$ (This is wrong. Let's assume the correct equation is based on supplementary - angle property). The angles $(15x - 52)$ and $(10x - 43)$ are supplementary (a linear pair), so $(15x - 52)+(10x - 43)=180$.

Step4: Combine like terms

$15x+10x-52 - 43=180$, which gives $25x-95 = 180$.

Step5: Solve for $x$

Add 95 to both sides: $25x-95 + 95=180 + 95$, so $25x=275$. Then divide both sides by 25: $x = 11$.

Step6: Find the value of $z$

Since $z$ and $(15x - 52)$ are vertical angles, substitute $x = 11$ into $15x-52$: $15\times11-52=165 - 52 = 113$. So $z = 113$.

Answer:

$x = 11$
$z = 113$