https://file.aiquickdraw.com/mathbot/file/1727390143854egyyaag3.jpg

Let's solve for \(x\) in the given problems.

### Problem 8:
We are given a triangle, and the sum of interior angles of any triangle is always \(180^\circ\). The angles are labeled as \(5x^\circ\), \(x^\circ\), and \(120^\circ\).

Using the triangle angle sum property:
\[
5x + x + 120 = 180
\]
Simplify:
\[
6x + 120 = 180
\]
Subtract 120 from both sides:
\[
6x = 60
\]
Divide by 6:
\[
x = 10
\]

### Problem 9:
In this problem, we have two angles formed by a straight line (a linear pair), so the sum of the two angles must be \(180^\circ\). The two angles are \(140^\circ\) and \(x^\circ + 35^\circ\).

Using the linear pair property:
\[
x + 35 + 140 = 180
\]
Simplify:
\[
x + 175 = 180
\]
Subtract 175 from both sides:
\[
x = 5
\]

### Problem 10:
This is a triangle where the sum of interior angles equals \(180^\circ\). The angles are labeled as \(x^\circ\), \(4x^\circ\), and \((3x + 54)^\circ\).

Using the triangle angle sum property:
\[
x + 4x + (3x + 54) = 180
\]
Simplify:
\[
8x + 54 = 180
\]
Subtract 54 from both sides:
\[
8x = 126
\]
Divide by 8:
\[
x = 15.75
\]

### Problem 11:
Another triangle, where the angles are labeled as \(x^\circ\), \((2x)^\circ\), and \((x - 20)^\circ\).

Using the triangle angle sum property:
\[
x + 2x + (x - 20) = 180
\]
Simplify:
\[
4x - 20 = 180
\]
Add 20 to both sides:
\[
4x = 200
\]
Divide by 4:
\[
x = 50
\]

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Let me know if you would like further details or explanations on any step.

### Related Questions:
1. How do you calculate the sum of interior angles for polygons with more than three sides?
2. What is the exterior angle theorem, and how is it useful in solving problems like these?
3. What is a supplementary angle, and where does it apply in geometry?
4. Can you explain how to find the angles of an isosceles triangle given one angle?
5. How does the Pythagorean theorem apply when solving triangles with right angles?

### Tip:
Remember that in a triangle, the exterior angle is always equal to the sum of the two non-adjacent interior angles.

For 8-12, solve for x in the given geometric diagrams involving triangles and straight angles.

This problem set involves solving for x in geometric diagrams, focusing on triangles and straight angles. Concepts covered include the Triangle Angle Sum Theorem and the Linear Pair Theorem. Suitable for middle school students (Grades 6-8).

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