Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![**Geometry Problem and Angle Calculation**
**Problem 5:**
In the given diagram of a circle with center \(N\):
- \(K\), \(L\), \(M\), and \(J\) are points on the circle.
- \(\angle JKL = ?\)
- \(\overline{JK} = 53^\circ\) (arc)
- \(\overline{KL} = 65^\circ\) (arc)
The diagram consists of a circle with center \(N\). The circle includes points \(K\), \(L\), \(M\), and \(J\) on its circumference.
Here is a step-by-step process on how to calculate the desired angle, \(m\angle JKL\):
1. **Sum of angles/arc measures in the circle:**
- Total 360 degrees in the circle.
- Known arcs: \(\overline{JK} = 53^\circ\), \(\overline{KL} = 65^\circ\).
- Hence, \(\overline{JL} = 53^\circ\) and \(\overline{LM} = 65^\circ\)
2. **Calculation:**
- To find \(m\angle JKL\), we use the property of angles subtended by arcs at the circumference:
\[
m\angle JKL = \frac{\text{arc } JL + \text{arc } KL}{2}
\]
Simplifying this will give us the value of the desired angle.
**Answer:**
- \(m\angle JKL = \frac{53^\circ + 65^\circ}{2}\)
- Solving this calculation will provide the final angle measurement.
This problem helps enhance understanding of geometric properties related to circles and angles. Such exercises are invaluable in improving students’ analytical and problem-solving skills in geometry.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F84979d05-3af2-4c8c-85a6-0d8920f2d2a2%2F15cc24fd-1e3d-4136-a4a6-e58fc639d785%2F9g33qqj_processed.jpeg&w=3840&q=75)
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