Question

δprt≅δmkt
p
k
y
65
40
t
z
x
r
m
given that δabc ≅δde
m<e = x = y =

Explanation:

Step1: Use angle - congruence property of congruent triangles

Since $\triangle PRT\cong\triangle MKT$, corresponding angles are equal. $\angle RPT=\angle KMT$, $\angle PRT=\angle MKT$, $\angle PTR=\angle KTM$.

Step2: Find the value of $x$

$\angle MKT = 65^{\circ}$, and $\angle PRT$ and $\angle MKT$ are corresponding angles. So $z = 65^{\circ}$.

Step3: Find the value of $y$

$\angle KMT$ and $\angle RPT$ are corresponding angles. In $\triangle PRT$, using the angle - sum property of a triangle ($\angle PTR = 40^{\circ}$, $\angle PRT=z = 65^{\circ}$), we know that the sum of angles in a triangle is $180^{\circ}$. In $\triangle MKT$, $\angle KMT=y$. In $\triangle PRT$, $y=180^{\circ}-(40^{\circ}+65^{\circ})$.
$y = 75^{\circ}$

Step4: Find the value of $x$

$\angle KMT$ and $\angle RPT$ are corresponding angles, so $x = 75^{\circ}$

Answer:

$x = 75^{\circ}$, $y = 75^{\circ}$, $z = 65^{\circ}$