10.2 Angles and segments in circles
2 min read•july 22, 2024
Circles are like geometric playgrounds, full of cool angle relationships. Central angles boss it from the middle, while inscribed angles hang out on the edge, measuring half as much as their central buddies.
The is a game-changer. It helps us figure out missing angles and shows how inscribed angles on the same arc are always twins. Plus, any angle in a is automatically a right angle!
Angles in Circles
Central and inscribed angles
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- Central angles have vertex at center of circle formed by two radii intercept an arc on
- Inscribed angles have vertex on circumference formed by two chords measure half of subtending same arc
- Circumscribed angles have vertex outside circle formed by two lines measure half difference of intercepted arcs
Inscribed angle theorem applications
- Theorem states measure of inscribed angle is half measure of central angle subtending same arc
- Can find measure of inscribed angle given central angle measure (if central angle is 120°, inscribed angle is 60°)
- Can determine measure of central angle given inscribed angle measure (if inscribed angle is 45°, central angle is 90°)
- Solve for unknown angles in problems involving inscribed and central angles (find missing angle measures in a circle with some given information)
Inscribed vs central angle relationships
- Inscribed angle measure always half measure of central angle subtending same arc
- All inscribed angles subtending same arc have equal measures (inscribed angles subtending same arc are congruent)
- Inscribed angles subtending semicircle are right angles measure 90° (angle inscribed in semicircle is a right angle)
Angles from chords and tangents
- Angles formed by two chords intersecting inside circle measure half sum of measures of arcs intercepted by angle and its vertical angle
- Angles formed by two lines intersecting outside circle measure half difference of measures of intercepted arcs
- Angles formed by secant and tangent intersecting outside circle measure half difference of measures of intercepted arcs
- Angles formed by two tangent lines intersecting outside circle measure half difference of measures of intercepted arcs
Key Terms to Review (17)
Angle Bisector Theorem: The Angle Bisector Theorem states that if a point lies on the bisector of an angle, then it is equidistant from the two sides of the angle. This theorem not only highlights the relationship between angles and distances but also plays a significant role in various geometric proofs and properties, such as triangle inequalities, relationships in circles, and establishing similarity in figures.
Arc Length: Arc length is the measure of the distance along a curved path defined by a section of a circle. It plays a crucial role in understanding how angles and segments interact within circles, as well as determining relationships between different parts of circles. By relating the arc length to the radius and the angle subtended at the center of the circle, one can derive important formulas that apply in various geometric contexts.
Central angle: A central angle is an angle whose vertex is located at the center of a circle, and its sides (or rays) extend to the circumference of the circle. This angle helps define important relationships within a circle, including its arcs and sectors, allowing for further understanding of angles, lengths, and areas related to circular geometry.
Circumference: Circumference is the distance around the edge of a circle, essentially serving as its perimeter. Understanding circumference is crucial for solving problems related to circles, as it directly connects to their area, equations, and the relationships between angles and segments. Furthermore, circumference plays an important role when discussing arc lengths and sector areas, making it a key term in comprehending the full nature of circular geometry.
Circumscribed Angle: A circumscribed angle is an angle formed by two chords in a circle which have a common endpoint, where the vertex of the angle is on the circle. This type of angle is directly related to the arcs that it subtends, creating important relationships between angles and arcs in circular geometry. The measure of a circumscribed angle is half the measure of the intercepted arc, making it a key concept when examining angles and segments in circles.
Congruence of Arcs: Congruence of arcs refers to the property that two arcs are considered congruent if they have the same measure and are subtended by the same angle at the center of a circle. This concept is crucial in understanding relationships between angles and segments within circles, particularly how they influence geometric constructions and proofs. Congruent arcs also play a significant role in establishing congruent circles and identifying properties related to angles formed by intersecting chords.
Inscribed Angle: An inscribed angle is formed by two chords in a circle that share an endpoint, known as the vertex, located on the circle itself. This angle intercepts an arc, which is the portion of the circle between the endpoints of the chords. The relationship between an inscribed angle and the intercepted arc is fundamental; specifically, the measure of the inscribed angle is always half that of the measure of the intercepted arc.
Inscribed Angle Theorem: The Inscribed Angle Theorem states that an inscribed angle in a circle is half the measure of the intercepted arc. This relationship highlights how angles and arcs interact within a circle, making it crucial for solving various geometric problems involving circles. Understanding this theorem allows for deeper insights into the properties of circles, including relationships between angles, arcs, and chords.
Measure of an arc: The measure of an arc is the angle formed at the center of a circle by two radii that connect the center to the endpoints of the arc. This measurement is expressed in degrees and directly relates to the length of the arc and the overall geometry of the circle. Understanding how to calculate and use the measure of an arc is essential for working with angles and segments in circles, including relationships with chords, sectors, and inscribed angles.
Radius: The radius is the distance from the center of a circle or sphere to any point on its boundary. This key measurement is essential in understanding the properties and formulas related to circles, spheres, and other three-dimensional figures, as it directly influences calculations for circumference, area, volume, and surface area.
Secant: A secant is a line that intersects a circle at two distinct points. This concept is crucial when discussing the properties of angles and segments in circles, as it helps in understanding relationships between angles formed by intersecting lines and the segments that are created. The secant line is also connected to the equations of circles, as it can provide key information about the distance between points and can aid in determining various properties of circles.
Secant Segment: A secant segment is a line segment that intersects a circle at two points. This term is crucial for understanding the relationships between angles and segments in circles, especially when it comes to determining lengths and solving geometric problems involving circles. Secant segments play a significant role in various circle theorems and can be used to derive important relationships, such as the power of a point theorem.
Sector: A sector is a portion of a circle defined by two radii and the arc between them. It represents the area enclosed by these two radii and the circle's boundary, making it an essential concept in understanding how angles and segments relate within circular geometry. The size of a sector can be expressed in degrees or radians, and it plays a critical role in calculating areas and arc lengths in circles.
Segment: A segment is a part of a line that is bounded by two distinct endpoints. In the context of geometry, segments are essential for understanding various concepts related to shapes and figures, especially when analyzing properties and relationships within circles. Segments can be used to describe chords, diameters, and radii, which are all critical in studying circles and their measurements.
Semicircle: A semicircle is half of a circle, formed by cutting a whole circle along its diameter. This shape has some unique properties, particularly in relation to angles and segments created by the diameter and any other chords or arcs within the circle. Semicircles are essential in understanding how angles relate to the arcs they intercept and how segments can be used to measure various aspects of circular geometry.
Tangent: A tangent is a straight line that touches a curve at a single point without crossing it. In relation to circles, the tangent line is perpendicular to the radius at the point of contact, creating important relationships with angles and distances associated with the circle.
Tangent Line: A tangent line is a straight line that touches a curve at exactly one point without crossing it. In the context of circles, the tangent line is perpendicular to the radius at the point of contact, showcasing a unique relationship between angles, segments, and the properties of circles. This concept is fundamental in understanding various geometric relationships and equations involving circles and lines.