Question

in the diagram, $\triangle jkl \cong \triangle qrs$. find $m\angle r$.

Explanation:

Step1: Match congruent angles

$\angle K \cong \angle Q$, so $3x + 3 = 30$

Step2: Solve for $x$

Subtract 3, divide by 3:
$3x = 30 - 3 = 27$
$x = \frac{27}{3} = 9$

Step3: Use right triangle angle sum

In $\triangle QRS$, $\angle S=90^\circ$, so $\angle R + \angle Q = 90^\circ$. Substitute $\angle R = 2y - 3x$, $\angle Q=30^\circ$:
$2y - 3x + 30 = 90$

Step4: Substitute $x=9$, solve for $y$

$2y - 3(9) = 90 - 30$
$2y - 27 = 60$
$2y = 60 + 27 = 87$
$y = \frac{87}{2} = 43.5$

Step5: Calculate $m\angle R$

Substitute $x=9, y=43.5$:
$m\angle R = 2(43.5) - 3(9)$

Answer:

$60^\circ$