Question

9 if $\triangle qrs \cong \triangle mnp$, $qr = 11$, $qs = 15$, $mp = 2x + 1$, $m\angle s = 84^\circ$, $m\angle r = 32^\circ$, and $m\angle m = (17y - 4)^\circ$. find the values of $x$ and $y$.

Explanation:

Step1: Use Congruent Triangles' Sides

Since $\triangle QRS \cong \triangle MNP$, corresponding sides are equal. So $QS = MP$. Given $QS = 15$ and $MP = 2x + 1$, we set up the equation:
$2x + 1 = 15$

Step2: Solve for x

Subtract 1 from both sides: $2x = 15 - 1 = 14$
Divide by 2: $x = \frac{14}{2} = 7$

Step3: Find Angle in $\triangle QRS$

In $\triangle QRS$, sum of angles is $180^\circ$. Given $\angle R = 32^\circ$, $\angle S = 84^\circ$, so $\angle Q = 180 - 32 - 84 = 64^\circ$

Step4: Use Congruent Triangles' Angles

Since $\triangle QRS \cong \triangle MNP$, $\angle M = \angle Q = 64^\circ$. Given $\angle M = (17y - 4)^\circ$, set up:
$17y - 4 = 64$

Step5: Solve for y

Add 4 to both sides: $17y = 64 + 4 = 68$
Divide by 17: $y = \frac{68}{17} = 4$

Answer:

$x = 7$, $y = 4$