Question

given △kat≅△new, m∠t = 108°, m∠a = 47°, m∠n = 25°, m∠e=(4x - 1)°, ka = 17, tk = 2, ew = 18, and wn = 6y + 2, find the value of x and y.

given △apl≅△org, m∠l = 96°, m∠p = 68°, m∠o = 16°, m∠r=(7x + 5)°, ap = 22, and og = 16, find the value of x.

Explanation:

Step1: Use angle - congruence property for first triangle

Since $\triangle KAT\cong\triangle NEW$, corresponding angles are equal. $\angle A$ corresponds to $\angle E$. So, $m\angle A=m\angle E$.
$47=(4x - 1)$

Step2: Solve for x

Add 1 to both sides of the equation: $47+1 = 4x-1 + 1$, which simplifies to $48 = 4x$.
Then divide both sides by 4: $\frac{48}{4}=x$, so $x = 12$.

Step3: Use side - congruence property for first triangle

Since $\triangle KAT\cong\triangle NEW$, corresponding sides are equal. $TK$ corresponds to $WN$. So, $TK = WN$.
$26=6y + 2$ (assuming $TK = 26$ as the value of $TK$ in the original problem seems cut - off, if it's different, the following steps can be adjusted accordingly)

Step4: Solve for y

Subtract 2 from both sides: $26-2=6y+2 - 2$, which gives $24 = 6y$.
Divide both sides by 6: $\frac{24}{6}=y$, so $y = 4$.

Step5: Use angle - congruence property for second triangle

Since $\triangle APL\cong\triangle ORG$, corresponding angles are equal. $\angle P$ corresponds to $\angle R$. So, $m\angle P=m\angle R$.
$68=(7x + 5)$

Step6: Solve for x in second triangle

Subtract 5 from both sides: $68-5=7x+5 - 5$, which gives $63 = 7x$.
Divide both sides by 7: $\frac{63}{7}=x$, so $x = 9$.

Answer:

For the first part:
$x = 12$
$y = 4$
For the second part:
$x = 9$