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.
Solve for x.
![### Geometry: Angles and Circles
#### Understanding Angles with a Circle
The diagram above presents a geometric scenario involving a circle and an angle formed by two lines intersecting outside the circle.
**Key features of the diagram:**
1. **Circle**: The primary shape in the diagram is a circle, with a center labeled "O".
2. **Radius**: A line segment is drawn from the center "O" to the point on the circle where it meets one of the intersecting lines.
3. **Angle inside the Circle**: An angle measuring 157° is formed between the radius and one of the intersecting lines.
4. **Lines Intersecting Outside the Circle**: Two lines intersect outside the circle, forming an external angle labeled as \( x^\circ \).
### Explanation:
In this configuration, the key idea is to understand how the external angle (\( x^\circ \)) is related to the internal angle (157°).
When you have two tangents intersecting outside of the circle, the measure of the external angle (\( x^\circ \)) can be determined by:
\[ x = \frac{1}{2} (180^\circ - \text{internal angle}) \]
In this specific case:
\[ x = \frac{1}{2} (180^\circ - 157^\circ) \]
\[ x = \frac{1}{2} \times 23^\circ \]
\[ x = 11.5^\circ \]
Thus, \( x^\circ \) is 11.5°. This type of geometry problem helps in understanding how angles relate to one another when dealing with circles, tangents, and points of intersection.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa61fad0-ceb7-416c-8279-8e32da5d69e0%2F64963eee-6385-4f55-8c0a-634fc729b625%2Fcfanp4_processed.png&w=3840&q=75)
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