Let's solve these problems step by step.

Triangle problems:

9.

• Triangle PQR is isosceles.
• $m\angle Q = 74^\circ$.
• In an isosceles triangle, two angles are equal, so $m\angle P = m\angle R$.
• Use the triangle angle sum property: $74 + 2x = 180$.
• Solve for $x$ (which represents $m\angle P = m\angle R$):
  $2x = 106$, $x = 53$.

10.

• Triangle EFG has $m\angle FEG = 106^\circ$.
• It's a right triangle, so use the triangle angle sum property: $90 + 106 + x = 180$.
• However, this means $x$, $m\angle G$, should not solve this, as one angle already exceeds 90. It's likely an error on the worksheet; otherwise, solve for $m\angle E = m\angle G = 37^\circ$.

11.

• Triangle ABC is an isosceles triangle.
• $m\angle B = m\angle C = 48^\circ$.
• Find $m\angle A$ using the triangle angle sum property: $48 + 48 + x = 180$, $x = 84$.

12.

• Triangle NMP is equilateral.
• All angles are equal, so $m\angle M = m\angle N = m\angle P = 60^\circ$.

Value of x Problems:

13.

• Angle expressions: $8x - 23^\circ$ and $34^\circ$.
• These angles form a straight line, so they add up to $180^\circ$: $8x - 23 + 34 = 180$ $8x + 11 = 180$ $8x = 169$ $x = 21.125$

14.

• For an isosceles triangle, set the expressions equal for two base angles or derive the total angle: $(9x - 3) + (9x - 3) + \text{vertex angle} = 180$
• Simplify accordingly and solve for $x$.

15.

• $\triangle ABCD$ is isosceles with $\angle D = \angle E$.
• Equal sides: $CD = 4x + 9$, $DE = 7x - 5$, $CE = 16x - 27$.
• Set $4x + 9 = 7x - 5$ to find $x$.
• Once $x$ is found, plug it back in to find lengths $CD, DE,$ and $CE$.

If you need further calculation or explanation on these steps, please let me know!