https://file.aiquickdraw.com/mathbot/file/1726033986888bhhlai03.jpg

Let’s solve both parts of Question 8 based on the image.

### Part (a):
We need to find the value of angle \( x \). From the diagram:
- Angle \( \angle AFG = 23^\circ \)
- Angle \( \angle GFE \) is the angle we are looking for (denoted as \( x \)).
  
From the diagram, we can see that \( \angle AFG \) and \( \angle GFE \) are on a straight line at point \( F \), meaning they are supplementary angles. 

So, we use the property of supplementary angles:
\[
x + 23^\circ = 180^\circ
\]
\[
x = 180^\circ - 23^\circ
\]
\[
x = 157^\circ
\]

Thus, for part (a), \( x = 157^\circ \).

### Part (b):
In this part, we need to find \( x \), the exterior angle of a triangle \( ABC \) at point \( B \). We know:
- \( \angle ABC = 98^\circ \) (the exterior angle),
- The two base angles \( \angle ACB = \angle BDC \), because they are opposite angles of an isosceles triangle.

For an isosceles triangle, the exterior angle is equal to the sum of the two interior opposite angles. So:
\[
\text{Exterior angle } x = \angle ACB + \angle BDC
\]
Since both base angles are equal, let each base angle be \( y \). Then:
\[
x = 2y
\]
Also, we know that the exterior angle is supplementary to \( 98^\circ \), i.e.,
\[
x + 98^\circ = 180^\circ
\]
\[
x = 180^\circ - 98^\circ = 82^\circ
\]

Thus, for part (b), \( x = 82^\circ \).

---

If you'd like more details or have any questions, feel free to ask! Here are some related questions you might want to explore:
1. What is the sum of angles in a triangle?
2. What are supplementary angles?
3. How can you determine angles in an isosceles triangle?
4. What is the relationship between exterior and interior angles of a triangle?
5. How can you calculate angles when lines are parallel?

**Tip:** Always check for supplementary and vertically opposite angles when solving for unknowns in geometry diagrams!

Find the value of the marked angles in each of the following diagrams: (a) angle at F is 23°, and (b) exterior angle at B is 98°.

Learn how to solve for unknown angles in a geometry problem involving supplementary angles and an isosceles triangle. This problem involves the calculation of an exterior angle of 98° and a supplementary angle of 23°, suitable for students in Grades 8-10.

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