Question

line segments $overline{pd}$, $overline{nf}$, and $overline{tx}$ intersect at point $f$ as shown where $mangle pfn = 90^{circ}$, $mangle xfp=(3x - 56)^{circ}$, $mangle nft=(4z + 7)^{circ}$ and $mangle tfd=(x + 2)^{circ}$. 6. find the value of $x$. 7. find the value of $z$. 8. what is $mangle nft$? 9. what is $mangle xfd$?

Explanation:

Step1: Find the value of x

Since $\angle XFP$ and $\angle TFD$ are vertical - angles, they are equal. So we set up the equation $3x - 56=x + 2$.
Subtract x from both sides: $3x-x-56=x - x+2$, which simplifies to $2x-56 = 2$.
Add 56 to both sides: $2x-56 + 56=2 + 56$, getting $2x=58$.
Divide both sides by 2: $x=\frac{58}{2}=29$.

Step2: Find the value of z

We know that $\angle PFN = 90^{\circ}$, and $\angle NFT+\angle TFD=90^{\circ}$. Since $x = 29$, then $\angle TFD=x + 2=29 + 2=31^{\circ}$.
We have the equation $(4z + 7)+31 = 90$.
First, simplify the left - hand side: $4z+38 = 90$.
Subtract 38 from both sides: $4z+38-38=90 - 38$, getting $4z=52$.
Divide both sides by 4: $z=\frac{52}{4}=13$.

Step3: Find $m\angle NFT$

Substitute $z = 13$ into the expression for $\angle NFT$. So $m\angle NFT=(4z + 7)^{\circ}=(4\times13 + 7)^{\circ}=(52 + 7)^{\circ}=59^{\circ}$.

Step4: Find $m\angle XFD$

$m\angle XFD=m\angle XFP+m\angle PFN$. We know $m\angle XFP=3x - 56$ and $x = 29$, so $m\angle XFP=3\times29-56=87 - 56 = 31^{\circ}$, and $m\angle PFN = 90^{\circ}$. Then $m\angle XFD=31^{\circ}+90^{\circ}=121^{\circ}$.

Answer:

1. $x = 29$
2. $z = 13$
3. $m\angle NFT = 59^{\circ}$
4. $m\angle XFD = 121^{\circ}$