To find the value of $x$, let’s analyze the problem using the angles formed by intersecting chords inside a circle.

The key rule here is: $\text{The measure of the angle formed by intersecting chords is equal to half the sum of the measures of the arcs intercepted by the angle and its vertical opposite.}$

Step 1: Identify the intercepted arcs

• The angle $x^\circ$ at point $K$ is formed by the intersection of chords $FG$ and $JH$.
• The arcs intercepted by the angle are:
  • Arc $FH$, which measures $123^\circ$,
  • Arc $GJ$, which measures $108^\circ$.

Step 2: Apply the formula

The formula for the angle is: $x = \frac{1}{2} \left( \text{measure of arc } FH + \text{measure of arc } GJ \right)$ Substitute the values: $x = \frac{1}{2} \left( 123 + 108 \right)$

Step 3: Solve

$x = \frac{1}{2} \cdot 231 = 115.5$

Final Answer:

$x = 115.5^\circ$

Would you like me to clarify anything further or provide additional examples?

Related Questions:

1. How does the property of intersecting chords relate to other circle theorems?
2. What happens if one of the intercepted arcs is unknown?
3. How is the rule applied when chords are extended outside the circle?
4. What are some practical applications of this theorem?
5. Can this property help find the arc measures if the angle is given?

Tip:

Always check whether the angles you calculate inside the circle are subtended by the correct arcs, as errors often arise from misinterpreting the diagram.