Question

given the figure below, find the values of x and z.

Explanation:

Step1: Use vertical - angles property

Vertical angles are equal. So, $14x - 28=6x + 8$.

Step2: Solve for x

Subtract $6x$ from both sides: $14x-6x - 28=6x-6x + 8$, which simplifies to $8x-28 = 8$. Then add 28 to both sides: $8x-28 + 28=8 + 28$, getting $8x=36$. Divide both sides by 8: $x=\frac{36}{8}=\frac{9}{2}=4.5$.

Step3: Find the value of one of the angles

Substitute $x = 4.5$ into $6x + 8$. We get $6\times4.5+8=27 + 8=35$.

Step4: Use the fact that the sum of angles around a point is 360°

Since the four angles around the intersection point sum to 360° and the vertical - angles are equal, we know that $2(14x - 28)+2z=360$. Substitute $x = 4.5$ into $14x - 28$ which is 35. Then $2\times35+2z=360$. So, $70+2z=360$. Subtract 70 from both sides: $2z=360 - 70=290$. Divide both sides by 2: $z = 145$.

Answer:

$x = 4.5$, $z = 145$