Identify the type of triangle.

The tick marks on sides NP and PO indicate that $NP = PO$. Therefore, $\triangle NPO$ is an isosceles triangle.

In an isosceles triangle, the angles opposite the equal sides are equal.

Thus, $x = 46^\circ$.

The sum of angles in $\triangle NPO$ is $180^\circ$.

The unknown values are:

In $\triangle RST$, $\angle R = 34^\circ$ and $\angle S = 72^\circ$.

The exterior angle $x$ at vertex T is equal to the sum of the two opposite interior angles.

The unknown value is:

The angle $y$ and the $67^\circ$ angle form a linear pair on a straight line. Angles on a straight line sum to $180^\circ$.

The sum of angles in $\triangle UTW$ is $180^\circ$.

Substitute the value of $y$:

The unknown values are:

We have two right-angled triangles: $\triangle HLN$ (right-angled at L) and $\triangle SRT$ (right-angled at R).
From the angle markings:
$\angle H = \angle S$ (one arc)
$\angle N = \angle T$ (two arcs)
$\angle L = \angle R = 90^\circ$ (right angle symbol)

Since all three corresponding angles are equal, the two triangles are similar by the AAA (Angle-Angle-Angle) similarity criterion.
The correspondence of vertices is $H \leftrightarrow S$, $L \leftrightarrow R$, $N \leftrightarrow T$.

For similar triangles, the ratio of corresponding sides is equal:

Given: $HL = 15$ cm, $HN = 18$ cm, $ST = 24$ cm, $SR = x$ cm.

Simplify the ratio on the right side:

So, the equation becomes:

Cross-multiply:

The length is in cm.
The unknown value is: