Question

given $\triangle phscong\triangle cnp$, find the values of $x$, $y$, and $z$.

Explanation:

Step1: Equate corresponding angles

Since $\triangle PHS\cong\triangle CNP$, corresponding angles are equal. $\angle H=\angle P$. So, $6x - 29=36$.

Step2: Solve for $x$

Add 29 to both sides of the equation $6x - 29=36$: $6x=36 + 29=65$. Then $x=\frac{65}{6}$.

Step3: Equate another pair of corresponding angles

$\angle S=\angle N$. So, $4z-32 = 115$.

Step4: Solve for $z$

Add 32 to both sides of the equation $4z-32 = 115$: $4z=115 + 32=147$. Then $z=\frac{147}{4}$.

Step5: Equate the third - pair of corresponding angles

$\angle P=\angle C$. Since the sum of angles in a triangle is $180^{\circ}$, in $\triangle PHS$, $\angle S=115^{\circ}$, $\angle P = 36^{\circ}$, then $\angle H=180-(115 + 36)=29^{\circ}$. In $\triangle CNP$, $\angle N = 115^{\circ}$, $\angle C=36^{\circ}$, and $\angle P=29^{\circ}$. Also, $3y-17=29$.

Step6: Solve for $y$

Add 17 to both sides of the equation $3y-17=29$: $3y=29 + 17=46$. Then $y=\frac{46}{3}$.

Answer:

$x=\frac{65}{6}$, $y=\frac{46}{3}$, $z=\frac{147}{4}$