On circle
\[O\] below, the measure of
\[\stackrel{\LARGE{\frown}}{FJ}\] is
\[84^\circ\]. The measure of
\[\stackrel{\LARGE{\frown}}{GH}\] is
\[76^\circ\].
What is the measure of
\[\angle HKJ\]?

\[^\circ\]
A circle centered at point O. Points F, G, H, and J all lie on the circle in a clockwise direction. Point K is in the circle to the right of the center of the circle, point O. A line segment connects point F to point G. A line segment connects point G to point J and goes through point K. A line segment connects point F to point H and goes through point K. Line segments F G, G K, and K F make triangle F G K. Angle H K J is made by line segment J K and line segment H K.

To solve for the measure of \(\angle HKJ\), we need to use the properties of circles and arcs. 

Given:
- The measure of the arc \(\stackrel{\LARGE{\frown}}{FJ}\) is \(84^\circ\).
- The measure of the arc \(\stackrel{\LARGE{\frown}}{GH}\) is \(76^\circ\).

Let's denote:
- The measure of the arc \(\stackrel{\LARGE{\frown}}{FH}\) by \(x^\circ\).
- The measure of the arc \(\stackrel{\LARGE{\frown}}{HJ}\) by \(y^\circ\).

Since the points \(F\), \(G\), \(H\), and \(J\) are on the circle, and the circle's total circumference is \(360^\circ\), we know that:

\[ 84^\circ + 76^\circ + x^\circ + y^\circ = 360^\circ \]

However, the precise measures of arcs \(\stackrel{\LARGE{\frown}}{FH}\) and \(\stackrel{\LARGE{\frown}}{HJ}\) are not necessary for finding the measure of \(\angle HKJ\).

To find \(\angle HKJ\):
1. Recognize that \(\angle HKJ\) is an interior angle of the circle.
2. According to the properties of circles, the measure of an interior angle is half the sum of the measures of the arcs intercepted by the angle.

The arcs intercepted by \(\angle HKJ\) are \(\stackrel{\LARGE{\frown}}{GH}\) and \(\stackrel{\LARGE{\frown}}{FJ}\).

Hence,

\[ \angle HKJ = \frac{1}{2} \left( \stackrel{\LARGE{\frown}}{GH} + \stackrel{\LARGE{\frown}}{FJ} \right) \]

Substitute the given arc measures:

\[ \angle HKJ = \frac{1}{2} \left( 76^\circ + 84^\circ \right) \]
\[ \angle HKJ = \frac{1}{2} \left( 160^\circ \right) \]
\[ \angle HKJ = 80^\circ \]

Therefore, the measure of \(\angle HKJ\) is \(80^\circ\).

Do you want more details or have any questions?

Here are 5 numbered questions you might want to ask:
1. How do you calculate the measure of other angles in a circle given arc measures?
2. What are the properties of interior angles in a circle?
3. How do you find the length of an arc given its measure and the radius of the circle?
4. Can you explain the difference between central angles and inscribed angles?
5. How do you find the area of a sector of a circle?

**Tip:** Remember that the sum of the measures of all arcs around a circle is always \(360^\circ\). This can help you determine unknown arc measures when you know some of them.

Learn how to calculate the angle HKJ on a circle with given arc measures of 84° and 76°. Understand the properties of circle arcs and interior angles using detailed step-by-step solutions. Suitable for high school students studying circle geometry.

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