Question

8 if △def≅△jkl, de = 18, ef = 23, df = 9x - 23, jl = 7x - 11, and jk = 3y - 21, find the values of x and y.
9 if △qrs≅△mnp, qr = 11, qs = 15, mp = 2x + 1, m∠s = 84°, m∠r = 32°, and m∠m=(17y - 4)°, find the values of x and y.

Explanation:

8.

Step1: Use congruent - side property

Since $\triangle DEF\cong\triangle JKL$, corresponding sides are equal. So $DF = JL$.
$9x - 23=7x - 11$

Step2: Solve for $x$

Subtract $7x$ from both sides: $9x-7x - 23=7x - 7x-11$.
$2x-23=-11$.
Add 23 to both sides: $2x-23 + 23=-11 + 23$.
$2x = 12$.
Divide both sides by 2: $x=\frac{12}{2}=6$.

Step3: Use another congruent - side property

Also, $DE = JK$. Since $DE = 18$ and $JK = 3y - 21$, we have $3y-21 = 18$.

Step4: Solve for $y$

Add 21 to both sides: $3y-21 + 21=18 + 21$.
$3y=39$.
Divide both sides by 3: $y=\frac{39}{3}=13$.

Step1: Use congruent - side property

Since $\triangle QRS\cong\triangle MNP$, corresponding sides are equal. So $QS = MP$.
$15=2x + 1$.

Step2: Solve for $x$

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

Step3: Use congruent - angle property

Corresponding angles are equal. So $m\angle Q=m\angle M$.
In $\triangle QRS$, $m\angle Q=180^{\circ}-m\angle R - m\angle S=180^{\circ}-32^{\circ}-84^{\circ}=64^{\circ}$.
Since $m\angle M=(17y - 4)^{\circ}$ and $m\angle M = m\angle Q = 64^{\circ}$, we have $17y-4 = 64$.

Step4: Solve for $y$

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

Answer:

$x = 6$, $y = 13$

9.