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!