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Notice that $\overline{QS}$ is a common side for triangles $PQS$ and $RSQ.$ By the Reflexive Property of Congruence, $\overline{QS}$ is congruent to itself. Additionally, $\angle PQS$ and $\angle RSQ$ are congruent, as are $\angle PSQ$ and $\angle RQS.$ Consequently, $△PQS$ and $△RSQ$ are congruent because of the Angle-Side-Angle (ASA) Congruence Theorem. Therefore, the corresponding sides and angles are congruent. By definition, congruent angles have the same measure, and congruent segments have the same length. Therefore, the congruence statements on the right-hand side support the formation of the following three equations. By solving Equation (I), the value of $z$ can be found. Next, solve Equation (III) to find the value of $x.$ Then, the value of $y$ can be found by substituting $x=1.5$ into the Equation (II) and solving the resulting equation for $y.$ Finally, the required sum can be calculated by substituting the values found for $x,$ $y,$ and $z.$

To find the value of $x,$ notice that $x,$ $m\angle DCB,$ $m\angle JCK,$ and $m\angle JCH$ add up $360^{\circ }.$ Since $R_1$ is a rectangle and $S_2$ is a square, $\angle DCB$ and $\angle JCH$ are right angles. Therefore, these angles have a measure of $90^{\circ }$ each. Also, it is given that $m\angle JCK=3z+8.$ Use this information to solve for $x.$ An expression of $x$ was found in terms of $z.$ Therefore, to find the value of $x,$ the value of $z$ should first be known.

Finding the Value of $z$

If $△DGL$ and $△CJK$ can be proven to be congruent, that would provide the needed information to find the value of $z.$ Therefore, focus on those two triangles.

Proving that $\overline{DL}\cong \overline{CK}$

Notice that $\overline{KL}$ is common to both triangles. By using the Segment Addition Postulate, the following pair of equations can be written. Since $\overline{DK}$ is congruent to $\overline{CL},$ these segments have the same length, that is, $DK=CL.$ Simlarly, $KL=LK.$ By substituting these expressions into Equation (I), a relation between $\overline{DL}$ and $\overline{CK}$ will be obtained. This equation implies that $\overline{DL}$ and $\overline{CK}$ are congruent.

Proving that $\overline{GL}\cong \overline{JK}$

Once more, the Segment Addition Postulate can be used to rewrite $GL$ and $JK.$ Because $△MKL$ is isosceles, $\overline{MK}$ is congruent to $\overline{ML}.$ Therefore, $ML=MK.$ Keep in mind that it is given that $\overline{GM}$ and $\overline{JM}$ are congruent. That means $GM=JM.$ These two expressions can be substituted into Equation (I). Based on the equation just obtained, it can be concluded that $\overline{GL}$ and $\overline{JK}$ are congruent.

Proving that $\overline{DG}\cong \overline{CJ}$

Since $R_1$ is a rectangle, $S_1$ and $S_2$ are squares, and $T_1$ and $T_2$ are isosceles triangles, the following consequences can be drawn.

Given	Consequence
$R_1$ is a rectangle	$\overline{DA}\cong \overline{CB}$
$S_1$ is a square	$\overline{DG}\cong \overline{DE}$
$S_2$ is a square	$\overline{CH}\cong \overline{CJ}$
$T_1$ is an isosceles triangle	$\overline{DE}\cong \overline{DA}$
$T_2$ is an isosceles triangle	$\overline{CB}\cong \overline{CH}$

Next, organize the information in the right-hand column in a flow chart and use the Transitive Property of Congruence to prove that $\overline{DG}\cong \overline{CJ}.$

Proving that $△DGL\cong △CJK$

Previously, the following three congruence statements were obtained. The Side-Side-Side (SSS) Congruence Theorem allows to conclude that $△DGL$ is congruent to $△CJK.$ Since corresponding parts of congruent triangles are congruent, it can be concluded that $\angle DGL$ is congruent to $\angle CJK.$ Therefore, $z=34.$

Finding the Value of $x$

Finally, to find the value of $x,$ substitute $z=34$ into the equation $x=172-3z.$ Then solve for $x.$