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.
![### Understanding Inscribed Angles and Minor Arcs
In this section, we'll explore the relationship between inscribed angles and the arcs they subtend. Consider the problem presented below:
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**Problem:**
In circle F with \( m \angle EHG = 30^\circ \), find the angle measure of minor arc \( \overset{\frown}{EG} \).
**Diagram:**
The diagram illustrates a circle centered at point F. Points E, H, and G lie on the circumference of the circle. A line segment is drawn from E to H and another from H to G, forming the inscribed angle \( \angle EHG \). The measure of \( \angle EHG \) is given as \( 30^\circ \).
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**Solution:**
To solve this problem, follow these steps:
1. **Inscribed Angle Theorem:**
The Inscribed Angle Theorem states that an inscribed angle is half the measure of the intercepted arc. Therefore, if \( \angle EHG = 30^\circ \), the arc it intercepts (in this case, arc \( \overset{\frown}{EG} \)) will be twice this measure.
2. **Calculate the Measure of Arc \( \overset{\frown}{EG} \):**
\[
\text{Measure of } \overset{\frown}{EG} = 2 \times m \angle EHG
\]
\[
\text{Measure of } \overset{\frown}{EG} = 2 \times 30^\circ = 60^\circ
\]
Thus, the measure of the minor arc \( \overset{\frown}{EG} \) is \( 60^\circ \).
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This concept is foundational in circle geometry and has wide applications in problems involving circles, angles, and arcs. Understanding the relationship between inscribed angles and their corresponding arcs enables us to solve similar problems efficiently.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1ee66a4-8e77-4cb9-b1b9-9bd40d747386%2F4af5a66d-614d-4206-bc51-b0e4fdad550a%2Fvy5vzo_processed.png&w=3840&q=75)
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