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Angles

This article is about angles in geometry. For other uses, see Angles (disambiguation). In geometry, an angle is the figure formed by two rays, line segments, or lines sharing a common endpoint called the vertex, which represents the inclination or rotation between them. This configuration quantifies the space or turn between the two elements, distinguishing angles from straight alignments. Angles are measured in units such as degrees or radians to indicate the extent of rotation; a full circle corresponds to 360 degrees or $2\pi radians, with one radian defined as the angle subtended by an arc equal to the radius of the circle. The degree system divides the circle into 360 equal parts, originating from ancient Babylonian astronomy, while radians provide a dimensionless measure based on arc length, preferred in advanced calculus and physics for their natural integration with circular functions. Based on their measures, angles are classified into several types: an acute angle measures less than 90 degrees, a exactly 90 degrees, an obtuse angle more than 90 but less than 180 degrees, a straight angle 180 degrees, and a reflex angle more than 180 but less than 360 degrees. Additional relationships include complementary angles, which sum to 90 degrees, and supplementary angles, which sum to 180 degrees, fundamental for solving geometric problems involving parallel lines and transversals. The concept of angles traces back to ancient Egyptian and for practical applications like and astronomy, with formalizing the definition in his around 300 BCE as the mutual inclination of two lines in a . Angles underpin key theorems in , such as the equality of vertically opposite angles and the sum of angles in a being 180 degrees, and extend to non-Euclidean geometries where such sums vary. In broader contexts, angles are essential in for defining sine, cosine, and functions, and in physics for analyzing forces, trajectories, and wave propagation.

Geometry and Mathematics

Definition and Basic Concepts

In geometry, foundational elements such as lines and rays provide the basis for understanding angles. A line is an infinite set of points extending in both directions, while a ray is a portion of a line that begins at a single endpoint, called the initial point or origin, and extends infinitely in one direction. These concepts are essential prerequisites, as angles are constructed from rays rather than full lines to capture the directional spread from a common starting point. An is the geometric figure formed by two distinct rays that share a common , known as the . The two rays are referred to as the sides or arms of the angle, and the figure encompasses the region between them. Formally, this union of rays defines the angle's structure, distinguishing it from a straight line, which would represent coincident rays. Angles are denoted using the ∠ followed by three letters, where the middle letter indicates the (e.g., ∠ABC, with B as the and rays BA and BC as sides), or by a single lowercase letter or number near the for simplicity. The basic components of an angle include the , the sides, and the interior and exterior regions. The interior of the angle consists of all points lying between the two sides, determined by selecting points on each side (other than the ) and including the points between them; this region forms the "opening" of the angle. Conversely, the exterior comprises the rest of the outside this interior region. In diagrams, angles are visually represented by the two rays emanating from the marked , often with a curved connecting points on the sides to highlight the interior, emphasizing the angular spread. Basic angles are typically undirected, focusing on the figure itself without specified , though directed angles may incorporate a of in advanced contexts.

Measurement and Units

Angles are quantified using standardized units that measure their magnitude relative to a full , which corresponds to a complete circle. The two primary units are and , each with distinct historical origins and rationales for their adoption. The , denoted by the °, divides a full circle into 360 equal parts, a convention tracing back to ancient Babylonian astronomers who employed a base-60 () number system and approximated the year as 360 days, facilitating astronomical calculations. This unit became widespread through adoption and persists in everyday and educational contexts due to its intuitive subdivision into (') and 60 seconds ("). In contrast, the , often denoted by "rad" or left without a , defines a full circle as exactly 2π , emerging from the need for a dimensionless measure in where the angle equals the of to on a . The concept was introduced by English mathematician in 1714 for trigonometric tables, though the term "radian" was coined later in 1873 by James Thomson in examination questions. Radians are preferred in advanced and physics for their compatibility with natural constants like π, avoiding conversion factors in derivations involving or series expansions. To convert between degrees and radians, the formula θ_rad = θ_deg × (π/180) is used, derived from the proportion of full-circle equivalents: since 360° = 2π radians, dividing both sides by 360 yields 1° = (2π)/360 = π/180 radians. For example, a 90° converts to (90) × (π/180) = π/2 radians, simplifying trigonometric identities and in polar coordinates. The inverse conversion is θ_deg = θ_rad × (180/π). An alternative unit, the (also called or gon, abbreviated "gon" or "grd"), divides a full circle into equal parts, proposed during the in the late as part of the system's decimal-based reforms to align angular measures with base-10 convenience. It finds niche applications in , , and , where right angles measure exactly 100 gradians, easing decimal computations for slopes and bearings. The conversion is 1 gon = 0.9°, or θ_gon = θ_deg × (10/9), since 400 gon = 360°. For precision, fractional angles are expressed as decimals or fractions with units: for instance, 30° or 0.5π radians (equivalent to π/6 radians). The degree symbol ° follows the number without space (e.g., 45°), while radians use "rad" explicitly in formal SI contexts or omit it when clear from context, as per international standards. Subdivisions like arcminutes and arcseconds apply mainly to degrees for high-precision work, such as astronomy.

Types and Classifications

Angles are primarily classified based on their measure in degrees, where a degree is a unit of angular measurement dividing a full circle into 360 equal parts. An acute angle measures less than 90 degrees, such as a 45-degree angle formed by the intersection of two lines in a typical coordinate plane diagram. A right angle measures exactly 90 degrees, often represented by the corner of a square. An obtuse angle measures greater than 90 degrees but less than 180 degrees. A straight angle measures exactly 180 degrees, aligning with a straight line. A reflex angle measures greater than 180 degrees but less than 360 degrees. A full rotation, or complete angle, measures exactly 360 degrees, corresponding to a complete turn around a point. Angles can also be classified by their relationships to one another, particularly in pairs formed by intersecting or . Adjacent angles share a common side and but do not overlap. Complementary angles are two angles whose measures sum to 90 degrees. Supplementary angles are two angles whose measures sum to 180 degrees. Vertical angles are formed when two lines intersect, consisting of the opposite angles at the intersection point, which are equal in measure. In the context of cut by a transversal, corresponding angles occupy the same relative position and are equal, while alternate angles—either interior or exterior—lie on opposite sides of the transversal and are also equal. Beyond planar classifications, angles in geometric configurations consider three-dimensional extensions. In three-dimensional contexts, spiral or helical angles describe the angular progression in curved paths, such as the constant angle between the to a and its axis, often parameterized in for structures like coils or screws.

Properties and Theorems

In , the sum of the interior angles of a is $180^\circ, a fundamental property established through the parallel postulate. To sketch the proof, draw a line parallel to one side of the triangle through the opposite ; the alternate interior angles formed with the transversal sides are equal, and the consecutive interior angles sum to $180^\circ, yielding the total angle sum. This property generalizes to polygons: the sum of interior angles in an n-sided is (n-2) \times 180^\circ. The derivation follows by triangulating the into n-2 triangles, each contributing $180^\circ, for a total of (n-2) \times 180^\circ. The states that an exterior angle of a equals the sum of the two remote interior angles. A proof sketch uses the fact that the exterior angle and the adjacent interior angle form a straight line summing to $180^\circ; subtracting the adjacent interior from this leaves the exterior equal to the sum of the other two interiors. The angle bisector theorem asserts that the bisector of an interior angle in a divides the opposite side in the ratio of the adjacent sides. For ABC with bisector from A meeting BC at D, BD/DC = AB/AC. A proof sketch constructs congruent using the angle bisector and applies the or area ratios in the sub-. In circle geometry, the inscribed angle theorem holds that an is half the subtending the same . Thus, if \angle BAC is inscribed and \angle BOC is central for arc BC, then \angle BOC = 2 \angle BAC. The proof sketch draws the radius to the inscribed angle's , forming isosceles triangles and using angles to show the inscribed angle as half the difference of arcs or directly half the central. Central angles thus measure the full arc, while inscribed angles measure half. For cut by a transversal, the alternate interior angles are equal, a consequence of the parallel postulate. The proof relies on assuming inequality leading to a contradiction with the postulate that such angles sum appropriately without intersection. These properties derive from Euclid's axioms, particularly the parallel postulate (fifth postulate), ensuring consistency in plane geometry without full axiomatic derivations here.

Applications and Contexts

In Trigonometry and Calculus

Trigonometric functions are fundamentally defined using ratios of sides in a . For an angle θ in a , the sine function is given by θ = / , the cosine by cos θ = adjacent / , and the tangent by θ = / adjacent. These definitions extend to the , where for an angle θ measured from the positive x-axis, θ equals the y-coordinate of the point on the , and cos θ equals the x-coordinate. This circular representation allows trigonometric functions to be defined for all real angles, capturing their periodic nature with period 2π. Key trigonometric identities arise from these definitions. The Pythagorean identity, sin² θ + cos² θ = 1, follows directly from the unit circle: the point (cos θ, sin θ) lies on the circle of radius 1, so by the , (cos θ)² + (sin θ)² = 1². To derive it from the , note that opposite² + adjacent² = hypotenuse²; dividing by hypotenuse² yields (opposite/hypotenuse)² + (adjacent/hypotenuse)² = 1, or sin² θ + cos² θ = 1. Angle addition formulas express trigonometric functions of sums or differences of angles. For sine, sin(A + B) = sin A cos B + cos A sin B; this can be derived geometrically using the unit circle and Ptolemy's theorem on the chord lengths, or via vector dot products of rotated unit vectors. Similarly, cos(A + B) = cos A cos B - sin A sin B follows from the same construction, with the cosine of the sum arising from the real part of the product of complex exponentials on the unit circle. These identities enable simplification of composite angles and underpin many trigonometric proofs. In , angles appear in rates of change and integrals. is the dθ/dt, representing the instantaneous rate of angular displacement with respect to time. Derivatives of rely on fundamental limits, such as lim_{θ→0} (sin θ)/θ = 1, which is established using the : for small θ > 0, sin θ < θ < tan θ in radians, and taking limits yields the result since cos θ → 1. This limit justifies d(sin θ)/dθ = cos θ and d(cos θ)/dθ = -sin θ at θ = 0, extending to all θ by the chain rule. Integration of trigonometric functions often involves their antiderivatives: ∫ sin θ dθ = -cos θ + C and ∫ cos θ dθ = sin θ + C, derived by reversing . More complex integrals, such as ∫ sin^m θ cos^n θ dθ, use formulas or substitutions based on identities like the Pythagorean, reducing powers until integrable. These techniques are essential for solving differential equations with periodic solutions. In , connects angles to exponentials: e^{iθ} = cos θ + i sin θ. This identity, introduced by Leonhard Euler in 1748, follows from the expansions: the series for e^{iθ} matches that of cos θ + i sin θ term by term. It unifies trigonometric and exponential functions, enabling representations like rotations as multiplications by e^{iθ} in the .

In Physics and Engineering

In physics, angles play a crucial role in describing rotational and directional phenomena, particularly in where they determine the effectiveness of . , a measure of rotational force, is calculated as the product of the force magnitude, the perpendicular distance from the axis of rotation, and the sine of the angle between the force and the position , given by the formula \tau = r F \sin \theta. This relationship highlights how the angle \theta maximizes torque at 90 degrees, influencing designs in levers and engines. In , the launch angle \theta directly affects the horizontal range R, expressed as R = \frac{v^2 \sin 2\theta}{g}, where v is the initial velocity and g is ; optimal range occurs at \theta = 45^\circ./3:_Two-Dimensional_Kinematics/3.3:_Projectile_Motion) Vector operations in physics rely heavily on angles to quantify interactions. The dot product of two vectors \mathbf{A} and \mathbf{B} is \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta, where \theta is the angle between them, providing a scalar measure of their alignment used in calculations like work done by a force./12:_Vectors_in_Space/12.03:_The_Dot_Product) Conversely, the magnitude of the cross product is |\mathbf{A} \times \mathbf{B}| = |\mathbf{A}| |\mathbf{B}| \sin \theta, representing the area of the parallelogram formed by the vectors and indicating their angular separation, essential for determining perpendicular components in magnetic fields or angular momentum./01:_Vectors_in_Euclidean_Space/1.04:_Cross_Product) In and wave physics, angles govern light and wave propagation across interfaces. describes as n_1 \sin \theta_1 = n_2 \sin \theta_2, where n_1 and n_2 are the refractive indices of the media, and \theta_1, \theta_2 are the angles of incidence and , respectively; this principle enables lens design and fiber by predicting beam bending. patterns arise from , with the angles of the minima determined by conditions like \theta \approx \frac{m \lambda}{d} for small angles in single-slit setups, where \lambda is , d is slit width, and m = \pm1, \pm2, \dots is the order, influencing resolution in microscopes and telescopes./University_Physics_III_-Optics_and_Modern_Physics(OpenStax)/04:_Diffraction/4.02:_Single-Slit_Diffraction) Engineering applications leverage angles for optimizing load distribution and motion . In bridge design, structural angles such as those in truss members distribute stress by aligning forces along inclined supports, reducing bending moments and enhancing stability under compressive and tensile loads, as seen in Warren truss configurations where 60-degree angles balance shear and axial forces. gears transmit between intersecting shafts at angles typically of 90 degrees, with the gear determined by the cone angles, allowing efficient power transfer in differentials and right-angle drives while minimizing slippage through precise angular meshing.

In Computer Graphics and Navigation

In , angles are fundamental to representing and orientations of objects in both and spaces. A common method for rotations involves the use of rotation matrices, which transform coordinates by an angle θ around the origin. The standard rotation matrix is given by: \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} This matrix rotates a point (x, y) to (x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta), enabling efficient computation of object orientations in rendering pipelines. In graphics, provide a straightforward way to describe an object's orientation through three sequential rotations about the x, y, and z axes, typically denoted as roll, pitch, and yaw. However, suffer from , a singularity where two axes align, reducing the effective from three to two and causing discontinuities in rotation paths. This issue is mitigated in practice by alternative representations like quaternions, though remain prevalent in user interfaces for their intuitiveness. Algorithms in computer graphics frequently leverage angles for dynamic simulations and rendering. Angle interpolation, or spherical linear interpolation (SLERP), is used in keyframe animations to smoothly transition between orientations, avoiding abrupt changes by interpolating along the shortest geodesic on the unit sphere. For instance, in character animation, SLERP ensures natural rotational motion by parameterizing quaternions derived from initial and final angles. In collision detection, ray casting employs angular fields of view to simulate visibility and intersection tests; rays are projected from a viewpoint within a defined angular cone (e.g., 90° horizontal field of view) to detect obstacles, optimizing computations in real-time environments like video games. In navigation systems, define directional and positional references essential for geospatial computations. Bearings represent directions as angles measured clockwise from , typically ranging from 0° to 360°, allowing precise course plotting in and applications. For long-distance routes on Earth's surface, calculates great-circle paths—the shortest routes between points—using angular distances between latitudes and longitudes. The , which computes the Δσ between two points (φ₁, λ₁) and (φ₂, λ₂), is widely implemented in GPS software: a = \sin^2\left(\frac{\Delta\phi}{2}\right) + \cos \phi_1 \cos \phi_2 \sin^2\left(\frac{\Delta\lambda}{2}\right), \quad c = 2 \atan2(\sqrt{a}, \sqrt{1-a}), \quad d = R c where R is Earth's radius, providing accurate distances without approximations for small angles. (Note: The original Sinnott reference is from Sky & Telescope, 1984; accessible via academic libraries.) Latitude and longitude serve as angular coordinates in GPS mapping, with latitude (φ) measured from the equator (0° to 90° north/south) and longitude (λ) from the prime meridian (0° to 180° east/west). These are converted to Cartesian (x, y, z) systems for 3D modeling via: x = R \cos \phi \cos \lambda, \quad y = R \cos \phi \sin \lambda, \quad z = R \sin \phi facilitating integration with graphics engines or flight simulators.

History and Etymology

Ancient Origins

The earliest conceptualizations of angles emerged in ancient and civilizations, primarily through practical applications in astronomy and construction. In , Babylonian astronomers employed a (base-60) system to divide the celestial into 360 parts, facilitating angular measurements for tracking planetary motions and the sun's path, a that persists in modern systems. In , surveyors constructed right angles—essential 90-degree inclinations—for aligning pyramid bases and boundaries using knotted ropes forming 3-4-5 triangles, a method leveraging the Pythagorean relation to ensure perpendicularity during the floods' annual resurveying. Greek mathematicians formalized angles axiomatically around the 6th to 3rd centuries BCE, building on empirical observations. , circa 585 BCE, demonstrated that an inscribed in a is a , using geometric insights from practices to prove properties of circles. Euclid's Elements, composed around 300 BCE, defined a plane as the inclination of two lines meeting in a plane but not forming a straight line, establishing it as a foundational concept in deductive without reliance on measurement. Non-Western traditions independently developed angular concepts for ritual and practical . In , the Sulba Sutras, Vedic texts dated to approximately 800–500 BCE, prescribed constructions for fire altars requiring precise right angles and approximations of circular arcs, such as transforming squares into circles while preserving area, using cord-based methods to form perpendiculars. In , the Nine Chapters on the Mathematical Art, compiled around 200 BCE during the , addressed right angles in its chapter through problems on right triangles and rectangular fields, applying hook-and-lute methods (similar to the ) for land measurement and engineering. The term "" derives from Latin angulus, meaning "corner" or "bend," a form tracing to *h₂eng- ("to bend" or "curve"), reflecting its origin in describing inclines or joints in physical structures.

Development in Modern Mathematics

In the 17th and 18th centuries, the development of fundamentally linked angles to algebraic representations, enabling precise calculations of angular measures through . , in his 1637 work , introduced coordinate where lines are defined by equations of the form y = mx + c, with the m representing the of the \theta that the line makes with the positive x-axis, given by \tan \theta = \frac{rise}{run} = m. This innovation transformed geometric problems involving angles into algebraic ones, laying the groundwork for and modern analysis. Building on this, Leonhard Euler advanced angular measurement in his 1748 treatise , where he defined assuming a of 1, implicitly establishing the as the natural unit of measure, with \theta in radians corresponding to equal to the . Euler's approach emphasized the 's convenience for infinite series expansions and differential equations, promoting its use over degrees in advanced . The saw the expansion of angle concepts beyond frameworks through non- geometries, challenging traditional notions of relations. Nikolai Lobachevsky's 1829 publication "On the Principles of Geometry" in the Messenger introduced , where the sum of angles in a triangle is less than \pi radians, and angles are analyzed using such as \tanh \phi for in hyperbolic contexts, distinct from Euclidean circular functions. This work demonstrated that parallel lines diverge, affecting deficits in polygons and enabling new theorems on asymptotic behaviors. Complementing this, Bernhard Riemann's 1854 habilitation lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" formalized , where triangle angle sums exceed \pi radians, with the excess \hat{A} + \hat{B} + \hat{C} - \pi proportional to the triangle's area via the , influencing on positively curved manifolds. These developments broadened angle theory to curved spaces, with Riemann's providing a framework for measuring angles invariantly. In the 20th century, angles integrated into abstract algebraic structures, particularly through vector spaces and . The inner product in linear algebra, formalized by in his 1844 Die Lineale Ausdehnungslehre and later refined by William Rowan Hamilton's quaternions (1843) and ' vector analysis (1880s), defines the angle \theta between vectors \mathbf{u} and \mathbf{v} via \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|}, enabling geometric interpretations in n-dimensional spaces and underpinning in Hilbert spaces. In fractal geometry, popularized by , who coined the term "" in 1975, and expanded in (1982), angles appear in self-similar constructions like the , where iterative 60-degree turns generate infinite perimeters with finite area, quantifying irregularity through angular branching ratios. Topological angles, such as the , emerged in , measuring how a closed encircles a point by the total angular change divided by $2\pi, as defined in early 20th-century works by , providing invariant characterizations of spatial embeddings without metric dependence. The standardization of angular units culminated in the mid-20th century with the (SI), adopted in 1960 by the 11th Conférence Générale des Poids et Mesures, designating the as the coherent derived unit for angles to facilitate dimensionless consistency in scientific equations. Initially classified as a supplementary unit alongside the for angles, the was reclassified as derived in 1995 by the 20th CGPM, reflecting its widespread preference in higher mathematics for natural alignment with exponential and trigonometric identities, as emphasized in Euler's legacy. This adoption promoted radians in international standards, enhancing precision in fields from physics to while preserving degrees for practical applications.

Other Uses

Ethnic and Historical Groups

The Angles were a Germanic tribe originating from the region known as , located in what is now in northern Germany and southern , during the CE. They belonged to the broader group of peoples, alongside tribes such as the and , and their homeland was characterized by coastal settlements and fishing communities. Following the withdrawal of Roman forces from Britain around 410 CE, the Angles began migrating across the , driven by population pressures, climate changes, and opportunities in the depopulated post-Roman provinces. Genetic and archaeological evidence supports a substantial influx of northern European ancestry into early medieval , aligning with these migrations starting in the mid-5th century. In alliance with the Saxons and Jutes, the Angles participated in the settlement and conquest of , establishing footholds in eastern and northern regions by the late . According to historical accounts like those of , they targeted areas such as and the Humber estuary, leading to the formation of powerful kingdoms including (by the early ) and Mercia (consolidated around 600 CE). These kingdoms emerged from a mix of military campaigns and gradual settlement, displacing or assimilating Romano-British populations, and by the , Angle rulers like those in had expanded their influence through warfare and diplomacy. The invasions were not a single event but a prolonged process spanning from approximately 450 to 650 CE, marked by conflicts such as the Battle of Mount Badon around 500 CE, after which Angle dominance grew in the north. The Angles played a pivotal role in shaping Anglo-Saxon culture, particularly through their dialect, which formed a major component of , the precursor to . Their language, a West Germanic variant, blended with Saxon and Jutish elements to create the linguistic foundation of , evident in early texts like the works of written in the . This cultural synthesis fostered a shared Anglo-Saxon identity, distinct from continental Germanic traditions, and the term "" derives etymologically from Engla land, meaning "land of the Angles," reflecting their prominence in naming the emerging nation. The tribe's name itself traces to Proto-Germanic angulaz, denoting a "hook" or "bend," a root shared with the geometric term "" via Indo-European origins related to . By the , the Angles had largely assimilated into the broader Anglo-Saxon society, losing distinct tribal identities amid Viking invasions and political unification under figures like . Archaeological evidence, such as the ship burial dated to around 625 CE in , illustrates this transition, featuring elite grave goods like a gilded and Byzantine silver that blend Angle craftsmanship with European influences, symbolizing the kingdom's wealth and cultural integration. This burial, associated with King , highlights the Angles' role in early medieval trade and artistry before their full merger into English identity.

Geographical Locations

Angles is a small in the department of the [Provence-Alpes-Côte d'Azur](/page/Provence-Alpes-Côte d'Azur) region in southeastern , with a population of 70 residents as of 2022. Located at approximately 43°57′N 6°33′E, the village lies in a mountainous area near the Verdon Regional Natural Park, contributing to the department's reputation for scenic lavender fields in nearby plateaus like . Another locality named Les Angles exists in the department of the Occitanie region, near the border with and part of the greater area, with a of 8,694 as of 2022. Situated at 43°57′N 4°46′E, this village features , including the 15th-century Église Notre-Dame-de-l'Assomption in primitive Gothic style with a prominent square tower. Beyond France, the Isle of Anglesey (Ynys Môn in Welsh) is an island and county in northwestern Wales, United Kingdom, with a population of approximately 69,100 as of 2024. Positioned at roughly 53°17′N 4°20′W, it holds significant Celtic historical sites, including prehistoric remains dating back to the Mesolithic period. Its name derives from Old Norse 'Ǫngullsey,' meaning 'hook island,' sharing an etymological root with the name of the ancient Angles tribe denoting a bend or hook. In the United States, Angle Lake is a minor body of water in Lake of the Woods County, , within the enclave—the northernmost point in the contiguous 48 states. Located near 49°21′N 95°04′W, this remote area is accessible primarily by boat or seasonal ice roads and is surrounded by . Many of these place names trace etymological roots to the word "angle," denoting a corner or bend in , such as the protruding in or the angular coastal features of ; others connect to historical tribal settlements by the Angles people.

Arts and Media

In music, the term "angles" has appeared in album titles and song concepts to evoke shifts in perspective or musical direction. ' fourth studio album, Angles, released in 2011, marked a departure from their earlier sound toward more expansive, synth-influenced , reflecting the band's internal creative tensions during production. Similarly, rapper Wale's 2021 single "Angles" featuring explores romantic viewpoints through and R&B, emphasizing multiple interpretive lenses in relationships. In literature, "angles" serves as a for oblique viewpoints and perceptual shifts, notably in Emily Dickinson's . Her poem "The Angle of a " (Fr 578) personifies a sliver of outdoor vista glimpsed through a curtain crack as a persistent, Venetian-like intruder on the speaker's gaze, symbolizing the intrusive nature of observation and the limits of domestic enclosure. Dickinson frequently employs "slant" or angular imagery to convey indirect truths, as in "Tell all the truth but tell it " (Fr 1263), where truth must approach circuitously to avoid overwhelming the "infirm Delight" of human , prioritizing gradual revelation over direct confrontation. In , such metaphorical angles manifest in narrative techniques that distort viewer alignment for emotional effect; the 2005 comedy Angels with Angles uses supernatural intervention to reframe personal regrets through humorous, otherworldly viewpoints. Visual arts have long incorporated physical angles to challenge spatial perception and conceptual depth. British-Indian sculptor Anish Kapoor's works, such as (2009), feature a tilted, mirrored steel panel that warps reflections based on the viewer's approach angle, blurring boundaries between object, environment, and observer in a manner that evokes and disorientation. His installations often demand multi-angular engagement to fully disclose form, as seen in pieces like Sectional Bodyworks (2000s series), where voids and protrusions alter dramatically with shifting sightlines, underscoring themes of absence and bodily fragmentation. In photography, the (or canted angle) tilts the camera horizon to induce tension and unease, a technique pioneered in German Expressionist films like (1920) and persisting in modern works to convey psychological imbalance. Modern media extends "angles" into interactive and auditory formats, fostering engagement through perceptual play. The mobile puzzle app Angle Jungle (2017) introduces geometric angles via jungle-themed challenges, educating users on and while building spatial reasoning skills for ages 6 and up. Similarly, Angleglyph Puzzle (released ) presents glyph-matching tasks that require aligning angular patterns to form designs, blending minimalist aesthetics with cognitive puzzles in a 2020s digital format. Podcasts like Exploring Every Angle (launched 2024) interview creatives on multifaceted life narratives, using "angle" to frame diverse approaches in and beyond. Episodes often dissect hidden viewpoints in personal and professional journeys, mirroring broader media trends toward layered, non-linear tales.