Let's go through each question in detail.

Problem 6: Find all missing angles

We have a triangle with the following angles provided:

• Angle 1 = 72°
• Angle 2 = 57°
• Angle 3 is unknown
• Angle 4 is unknown

Since the sum of angles in a triangle is 180°, we can find the missing angle (Angle 3) by calculating: $m\angle 3 = 180^\circ - 72^\circ - 57^\circ$ Let's calculate that.

Calculations:

$m\angle 3 = 180^\circ - 72^\circ - 57^\circ = 51^\circ$

For Angle 4, assuming it's an exterior angle and based on the properties of angles, it would be: $m\angle 4 = 180^\circ - m\angle 1 = 180^\circ - 72^\circ = 108^\circ$

Problem 7: Find all missing angles in a complex figure

To solve this, we need to apply angle relationships like vertical angles and supplementary angles. Since we don't have the exact setup of each angle's relationship here, let's focus on the immediate angle relationships that may be present.

1. If $\angle 1$ is 129°, and assuming there are adjacent angles or vertical angles, we might use those relationships.
2. Use supplementary angles if they add up to 180° with nearby angles.

Let's solve based on your preference or additional clarification you may have on the figure.

Problem 8 and 9: Solve for $x$

1. Problem 8:

   • This looks like a pair of triangles with expressions for their side lengths in terms of $x$.
   • Use the equality of side lengths or any given relationships to form an equation for $x$.

2. Problem 9:

   • Similar approach: Equate the angles in terms of $x$ and solve accordingly.

Please confirm if you’d like a detailed solution for each question, especially for complex angle relationships in Problem 7.

Would you like additional details or have any specific questions?

Here are five related questions to expand on these concepts:

1. How do we determine missing angles in any polygon using the sum of interior angles?
2. What are vertical angles, and how are they used in angle calculations?
3. Can supplementary and complementary angles help in more complex shapes beyond triangles?
4. How does one use exterior angles to find unknown interior angles in a polygon?
5. What are some shortcuts to solving equations with geometric expressions?

Tip: Always start with known angle relationships (like sums and vertical angles) to simplify complex diagrams.