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.
![**Problem Statement:**
In circle \( L \) with \( m \angle KNM = 53^\circ \), find the angle measure of minor arc \( KM \).
**Diagram Explanation:**
The provided diagram includes a circle with center \( L \). Points \( K \), \( N \), and \( M \) are marked on the circumference. The line segment \( KN \) and line segment \( NM \) form an angle \( \angle KNM = 53^\circ \) at point \( N \). The arc \( KM \) connects points \( K \) and \( M \) on the circle. We are to find the angle measure of the minor arc \( KM \).
**Solution Section:**
To determine the angle measure of minor arc \( KM \), recall that in a circle, the measure of the arc is related to the measure of the central angle that intercepts the arc. Since the angle \( \angle KNM \) is given as \( 53^\circ \), it is an inscribed angle for arc \( KM \). The measure of an inscribed angle is always half the measure of the intercepted arc. Therefore, the measure of the minor arc \( KM \) can be calculated using the following relationship:
\[ m \text{(minor arc } KM) = 2 \times m \angle KNM \]
Given:
\[ m \angle KNM = 53^\circ \]
Substituting the given value,
\[ m \text{(minor arc } KM) = 2 \times 53^\circ \]
\[ m \text{(minor arc } KM) = 106^\circ \]
**Answer:**
The angle measure of minor arc \( KM \) is \( 106^\circ \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9ceab91b-0528-4b5e-b4aa-14fac5cac61d%2F960d4350-894d-4b93-9392-114f8dfe9cb3%2Fp66p2a9_processed.jpeg&w=3840&q=75)
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