Question

1. if △def≅△gjh, find each missing measure.

\\(\overline{df}=\\)
\\(\overline{jh}=\\)
\\(\overline{gj}=\\)
\\(m\angle f =\\)
\\(m\angle g =\\)
\\(m\angle j =\\)

1. given △rws≅△tuv, find the values of \\(x\\) and \\(y\\).

\\(x =\\)
\\(y =\\)

Explanation:

Step1: Recall property of congruent triangles

Corresponding sides and angles of congruent triangles are equal.

Step2: Find corresponding sides for $\triangle DEF\cong\triangle GJH$

Since $\triangle DEF\cong\triangle GJH$, $DF = GH$, $JH=EF$, $GJ = DE$. Given $DE = 19m$, $EF = 14m$, $GH = 16m$. So $DF=16m$, $JH = 14m$, $GJ=19m$.

Step3: Find corresponding angles for $\triangle DEF\cong\triangle GJH$

$\angle F=\angle H$, $\angle G=\angle D$, $\angle J=\angle E$. Given $\angle D = 34^{\circ}$, $\angle H=105^{\circ}$. So $m\angle F = 105^{\circ}$, $m\angle G=34^{\circ}$, and since the sum of angles in a triangle is $180^{\circ}$ in $\triangle DEF$, $\angle E=180-(34 + 105)=41^{\circ}$, so $m\angle J = 41^{\circ}$.

Step4: For $\triangle RWS\cong\triangle TUV$

Corresponding angles are equal. So $\angle R=\angle T$, then $8x - 27=29$. Solving for $x$:
$8x=29 + 27$, $8x=56$, $x = 7$.
Corresponding sides are equal. So $WS=UV$, then $20=3y + 7$. Solving for $y$:
$3y=20 - 7$, $3y = 13$, $y=\frac{13}{3}$.

Answer:

$DF = 16m$
$JH = 14m$
$GJ=19m$
$m\angle F = 105^{\circ}$
$m\angle G=34^{\circ}$
$m\angle J = 41^{\circ}$
$x = 7$
$y=\frac{13}{3}$