https://file.aiquickdraw.com/mathbot/file/1726469785112u9cjl19z.jpg

Let's go through each part of **Question 5** to determine the value of \( x \).

### Part (a)
We are given two angles, \( 30^\circ \) and \( 45^\circ \), along with an unknown angle \( x \). These angles appear to be part of a linear pair, which means that the sum of these angles is equal to \( 180^\circ \).

The equation for this situation will be:
\[
x + 30^\circ + 45^\circ = 180^\circ
\]
Simplifying:
\[
x + 75^\circ = 180^\circ
\]
\[
x = 180^\circ - 75^\circ = 105^\circ
\]

Thus, the value of \( x \) in part (a) is \( 105^\circ \).

### Part (b)
In this part, we are given four angles related to a set of intersecting lines. One of the angles is labeled as \( 95^\circ \), another as \( x + 10^\circ \), a third as \( 2x + 20^\circ \), and the fourth is vertically opposite the angle \( x + 10^\circ \).

Since vertically opposite angles are equal, we can set up the equation:
\[
x + 10^\circ = 95^\circ
\]
Solving for \( x \):
\[
x = 95^\circ - 10^\circ = 85^\circ
\]

Thus, the value of \( x \) in part (b) is \( 85^\circ \).

### Part (c)
This part involves a triangle with one angle given as \( 72^\circ \) and another as \( x \). The third angle is unknown, but we know that the sum of the interior angles of a triangle is always \( 180^\circ \).

Let's denote the third angle as \( 72^\circ \) (since it appears the same in the image), and set up the equation:
\[
x + 72^\circ + 72^\circ = 180^\circ
\]
Simplifying:
\[
x + 144^\circ = 180^\circ
\]
\[
x = 180^\circ - 144^\circ = 36^\circ
\]

Thus, the value of \( x \) in part (c) is \( 36^\circ \).

---

Would you like a further breakdown of any part of the solutions?

Here are 5 follow-up questions related to the concepts in this problem:

1. How do you determine the value of angles in a linear pair?
2. Why are vertically opposite angles always equal?
3. Can the sum of interior angles in any polygon be calculated similarly to triangles?
4. How would you solve for unknown angles in a polygon with more than three sides?
5. What is the significance of supplementary angles, and how are they different from complementary angles?

**Tip:** When solving geometry problems, always look for relationships like linear pairs, vertically opposite angles, and angle sums in polygons. These help break down complex diagrams into solvable parts.

Determine the value of x in each part of the problem, including angles in a linear pair, intersecting lines, and a triangle.

Learn how to solve for unknown angles using concepts such as linear pairs, vertically opposite angles, and the sum of interior angles in a triangle. This detailed explanation covers problems suitable for students in Grades 8-10.

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