I can help analyze and solve the problems in the image. Let me extract the necessary details.

Problem Breakdown:

1. Top Left Triangle ($\triangle PQR$):

   • Given: Angle $Q = 51^\circ$, two equal sides ($PQ = QR$).
   • This is an isosceles triangle, so the base angles ($\angle P$ and $\angle R$) are equal.

2. Top Right Triangle ($\triangle STU$):

   • Given: $\angle T = 68^\circ$, $ST = TU$ (isosceles triangle).
   • Find the unknown angles $\angle S$ and $\angle U$.

3. Bottom Triangle ($\triangle RST$):

   • Given: $\angle R = (9x + 4)^\circ$, $\angle S = (3x - 2)^\circ$, and $\angle T$ (vertex opposite equal sides).
   • Use the triangle angle sum property ($\sum \text{angles} = 180^\circ$) to find $x$ and all angles.

Solving Each Triangle:

1. $\triangle PQR$:

$\text{Sum of angles in a triangle} = 180^\circ.$ Let $\angle P = \angle R = x$. $x + x + 51 = 180 \implies 2x = 129 \implies x = 64.5.$

• $\angle P = \angle R = 64.5^\circ$.

2. $\triangle STU$:

$\text{Let} \, \angle S = \angle U = x \, \text{(isosceles triangle)}.$ $x + x + 68 = 180 \implies 2x = 112 \implies x = 56.$

• $\angle S = \angle U = 56^\circ$.

3. $\triangle RST$:

Using the triangle angle sum property: $(9x + 4) + (3x - 2) + \angle T = 180.$ Simplify: $12x + 2 + \angle T = 180 \implies \angle T = 180 - 12x - 2 \implies \angle T = 178 - 12x.$

• Now substitute the equality of the two sides ($RT = ST$) into the equations: This implies the base angles ($\angle R$ and $\angle T$) are equal.

Set $(9x + 4) = (178 - 12x)$: $9x + 4 = 178 - 12x \implies 21x = 174 \implies x = 8.29 \, (\text{rounded to 2 decimal places}).$

Substitute $x = 8.29$ back to find angles: $\angle R = 9(8.29) + 4 = 78.61^\circ,$ $\angle S = 3(8.29) - 2 = 23.87^\circ,$ $\angle T = 178 - 12(8.29) = 78.61^\circ.$

Final Answers:

1. $\triangle PQR$:

   • $\angle P = \angle R = 64.5^\circ$, $\angle Q = 51^\circ$.

2. $\triangle STU$:

   • $\angle S = \angle U = 56^\circ$, $\angle T = 68^\circ$.

3. $\triangle RST$:

   • $x = 8.29$,
   • $\angle R = 78.61^\circ$,
   • $\angle S = 23.87^\circ$,
   • $\angle T = 78.61^\circ$.

Would you like detailed steps for any part?
Here are 5 related questions to expand on this topic:

1. How do you identify an isosceles triangle and its properties?
2. Can we verify the solutions by checking the sum of angles in a triangle?
3. What happens if a triangle has obtuse angles? How does it affect the base angles?
4. How do you solve problems involving external angles of a triangle?
5. Could the method differ if the triangle was scalene instead of isosceles?

Tip: When working with isosceles triangles, always check the given equal sides to set the equal angles correctly!