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 \[\angle HKJ\]?**

Given that we have the measure of \(\stackrel{\LARGE{\frown}}{FJ}\) as \(84^\circ\) and the measure of \(\stackrel{\LARGE{\frown}}{GH}\) as \(76^\circ\), let's determine the measure of \(\angle HKJ\) using the properties of circles.

**Key Points:**
1. The measure of an interior angle of a circle is half the sum of the measures of the arcs intercepted by the angle.
2. \(\angle HKJ\) intercepts the arcs \(\stackrel{\LARGE{\frown}}{GH}\) and \(\stackrel{\LARGE{\frown}}{FJ}\).

### Step-by-Step Solution:

1. **Identify the Intercepted Arcs:**
   - \(\angle HKJ\) intercepts arc \(\stackrel{\LARGE{\frown}}{GH}\) with measure \(76^\circ\).
   - \(\angle HKJ\) also intercepts arc \(\stackrel{\LARGE{\frown}}{FJ}\) with measure \(84^\circ\).

2. **Sum of the Measures of the Intercepted Arcs:**
   - The sum of these intercepted arcs is:
     \[
     76^\circ + 84^\circ = 160^\circ
     \]

3. **Measure of the Interior Angle:**
   - According to the property of circles, the measure of an interior angle is half the sum of the measures of the intercepted arcs:
     \[
     \angle HKJ = \frac{1}{2} \left( 76^\circ + 84^\circ \right) = \frac{1}{2} \left( 160^\circ \right) = 80^\circ
     \]

### Final Answer:
The measure of \(\angle HKJ\) is \(80^\circ\).

Would you like 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:** Always remember that the measure of an angle formed by two intersecting chords is half the sum of the measures of the arcs intercepted by the angle. This is useful in solving many circle-related problems.

Discover how to find the measure of angle HKJ in circle O using arc measures of 84° and 76°. This problem involves circle geometry and the properties of interior angles. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation