Lines and Angles

Lines and angles are one of the basic concepts in geometry. It is important to understand these concepts to build other more complex ideas.

A line is a straight one-dimensional figure that extends infinitely in both directions. An angle is a figure formed by two rays with a common endpoint, called the vertex.

Lines

Lines and angles are one of the most important elements that form the foundation for geometry. A line is a figure that extends infinitely in both directions, and an angle is a shape formed when two lines intersect. There are different types of lines and angles, such as parallel lines and perpendicular lines, and each has its own set of properties. The various properties of lines and angles are explored in this lesson through the use of real-life examples and logical reasoning.

A line is made up of an endless number of closely spaced points. It has no width or depth, and is a one-dimensional figure. It is represented on a plane by the linear equation ax + by = c, where a and b are constants.

When two straight lines intersect in a plane, the angle formed is called an angle of intersection. The two lines that make up the angle are known as its arms (sides), and the point where they meet is called the vertex. An angle is measured in degrees, which range from 0° to 360°.

There are many different types of lines and line segments in a plane. They can be categorized into several kinds based on their positioning with respect to each other. Two lines can be parallel if they never meet each other, or intersect at the same point. Lines can also be intersecting if they meet at distinct points, or transversal lines, which cut two or more lines at distinct points.

There are also many types of angles that can be formed between two lines. Adjacent angles are those that have a common side and a common vertex. For example, the angles OA and BOA are adjacent, since they have both a common arm and a common vertex. Vertically opposite angles are also formed when two lines intersect at a common point, or vertex. For example, the angles AB and AC are vertically opposite.

Angles

Among the most important geometry concepts are lines and angles. Having a good understanding of how angles relate to one another can help you solve tricky questions on the GMAT, GRE and other competitive exams. Knowing the different types of angles and their relationships is also essential in understanding many other geometry concepts, such as triangles.

Two lines can intersect at any point and form an angle. This angle can be measured in degrees, with a full circle being 360°. In geometry, there are different types of angles, such as acute and obtuse. Each type of angle has its own name and characteristics. For example, an acute angle has a sharp point while an obtuse angle has a rounded point.

In addition to parallel and perpendicular lines, there are other types of line segments and angles. A line segment is a pair of end points with a definite length that extends between them. An angle is a figure formed by two rays that meet at a common point, called the vertex.

Whether an angle is congruent or not depends on how its vertices are located in relation to each other. If the vertices are on the same side of the line, they are congruent; if they are on opposite sides of the line, they are non-congruent. Additionally, the measure of an angle can be based on the number of rays that meet at its vertex.

When a pair of angles share a relationship with each other, they are referred to as related angles. For example, a pair of interior angles on the same side of a transversal are supplementary, while a pair of alternate interior angles is not. Additionally, the measure of an exterior angle is a combination of its measure and that of a corresponding interior angle.

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A Beginner’s Guide to Geometry

Geometry is a branch of mathematics that studies the properties of shapes. It includes planes, lines, angles and solids. It also deals with the relationships between them.

Children need to learn the names of the geometric shapes, as they will use them in many problems. A line is straight if it extends in both directions without end (infinitely). An angle is a figure with two rays sharing a common point called the vertex.

Planes

The Greek mathematician Euclid developed the concept of planes, along with a number of other fundamental concepts, which form the basis for all geometry that comes after him. All two-dimensional geometry is based on the idea of a plane, which is flat and extends infinitely into space without any height or width.

As a noun, “plane” can mean an airplane or a tool for smoothing wood, but it’s also used in mathematics to refer to any flat figure. Renaissance artists also helped lay the groundwork for plane geometry through their techniques of drawing in perspective.

Any three points that don’t fall on a single line are on one plane, and any two intersecting planes are always parallel to each other. The XY plane, shown below, is an example of an intersecting plane.

Polygons

In geometry, a polygon is a flat, plane, closed shape with straight sides that do not intersect each other. The angles of a polygon must be equal and measure less than 180°. A regular polygon is a convex figure.

Each side of a polygon connects to two or more vertices. A polygon can have a variety of sides, but the sides must be non-intersecting. A triangle is a simple polygon with three sides. A quadrilateral is a four-sided polygon, a pentagon has five sides, and a hexagon has six sides.

Each interior angle of a regular polygon is equal to 180°, and each exterior angle is 360°. Polygons have different names depending on the number of sides they have. If a polygon has n sides, it is called an n-gon.

Angles

Angles are a key concept in geometry. They help us build better streets and cities, tell time using shadows and sunlight, and even understand our solar system.

There are two main parts to an angle: the vertex and the arms. The vertex is the corner point of an angle, where two rays meet. The arms are the two sides of an angle, joined at a common endpoint. For example, in the given figure, OA and OB are the two arms of the angle.

Adjacent angles are those that share a vertex and a side but don’t overlap with one another. Supplementary angles are those whose sum is equal to 180 degrees. Alternate exterior angles are pairs of angles that appear on the outsides of parallel lines and on opposite sides of a transversal.

Lines

Lines are one-dimensional figures that are straight and extend forever in opposite directions. They are often seen in many 2D shapes and real world items. A line is made up of many different parts called vertices.

Students will first learn about lines as part of geometry in 4th grade. They will learn about the types of lines and how they are classified. They will also learn about the relationship between a line, a line segment and a ray.

It is important to understand the difference between a line and a line segment. A line has no endpoints, while a line segment has a definite beginning and an end. You may see the latter in a length of string or on a piece of paper.

Circles

In geometry, a circle is a 2D shape which has a radius. It is a locus of points that are equidistant from a common point (the centre of the circle).

Two circles are called congruent if their centres are at the same distance apart. Equal chords of a circle are always equidistant from the centre and the perpendicular bisector of a chord passes through the centre of the circle.

A circle can be split into parts based on their position. A segment of a circle created by a chord is also known as an arc. It can be divided into major and minor arcs. A sector of a circle is the area enclosed by two radii and their corresponding arcs. It can be divided into two types – minor sector and major sector.

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The Basics of Geometry

Geometry is a significant branch of math that involves shapes, sizes, diagrams and angles. It has many practical applications. Its roots can be traced to ancient civilizations.

The ancient Greek mathematician Euclid coined the term geometry and systematically arranged the subject under a set of postulates and theorems that we still follow today.

Points

The field of geometry studies points, lines and planes in two dimensions. Geometry also includes the study of three-dimensional solid shapes like spheres, cubes, cones, prisms and pyramids.

The discipline of geometry, along with arithmetic, is one of the oldest branches of mathematics. Its development dates back to ancient times, with Euclid’s logical system codified in his Elements around 300 bce.

Modern geometry encompasses many subfields including analytic and synthetic geometry, differential and algebraic geometry. It also explores the properties of curved surfaces in spherical and hyperbolic geometry. Mensuration in geometry includes the calculation of perimeter, area and capacity of plane shapes, and the volume of solids.

Lines

A line is a one-dimensional geometric figure with length but not width. It extends infinitely in either direction and has no endpoints. There are many different types of lines. Some are straight and others are curved.

Two lines that are perpendicular to each other are called parallel lines. The perpendicularity of lines is very important in geometry, as it allows for the construction of right angles. Many geometric operations, such as finding the slope of a line, depend on this property.

A line segment is a section of a line that has a definite starting point and ending point. The points on the line that are at the same distance from the starting point are called collinear points.

Planes

A plane is a flat surface that has one dimension, length. Any two non-collinear points that are not parallel to each other must be on separate planes. They can intersect along a line.

Points, also known as points of coordinates, are a fundamental concept in geometry. They are represented by a dot and are named using capital letters. Points do not have height or width, so they are non-dimensional.

A plane is a mathematical space that has both affine properties and metrical properties induced by a coordinate system. Points in the plane can be specified uniquely by their ordered pair of coordinates, which are the signed distances from a point to the perpendicular lines that intersect it.

Angles

Angle measurement is a critical concept that engineers use to construct buildings, bridges, houses etc. Athletes and artists also use this concept to enhance their performance.

In geometry, an angle is a figure formed by two rays that share a common endpoint (also called the vertex of the angle). There are 7 types of angles: Acute angle, Obtuse angle, Right Angle, Reflex angle, Complete angle and Straight angle.

Each of these angles has a set of measurements that are unique to that type of angle. You can identify an angle by its measurements and name it based on its shape and direction. This knowledge is required for all geometry questions, including those in competitive exams like GMAT and CAT.

Polygons

Children will extend their knowledge of multi-sided shapes by learning about polygons. A polygon is a flat two-dimensional shape with straight sides that are connected to form a closed shape and have vertices (corners).

Children learn about regular and irregular polygons, including equilateral triangles and squares. They will use their knowledge of angles and sides to distinguish between the different types of polygons.

They will identify that a regular polygon has equal sides and angles, while an irregular polygon does not. They will also be taught how to find the perimeter of a polygon based on its number of sides. They will then learn about the interior and exterior angles of a polygon.

Circles

In geometry circles are closed curves with all points equidistant from a special point called the center. A circle has a lot of properties that make it interesting and unique.

One property is that tangents to a circle touch it at just one point. Two tangents drawn from the same point to the circle at different angles are equal (AM = MB).

A line segment that joins two points on the circumference of a circle is called a chord. Not every chord is a diameter, but every diameter satisfies the definition of a chord. Angles subtended by a chord are called inscribed angles. A circular region bounded by two radii is a sector.

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FAQs

Q1. What is 스포츠중계?
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Q2. When did 스포츠중계 begin?
A2. The initial televised sports event, a college baseball match, aired in 1939.

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FAQs

Q1) What is NBA중계?
A1) NBA중계 refers to the live broadcast of NBA games, which includes pre-game insights, live-action commentary, and post-game analysis.

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