## Geometry

“Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.”

― Johannes Kepler

*Fundamental Concept on Geometry *

*Points*

*We may think of a point as a “dot” on a piece of paper or the pinpoint on a board. In geometry, we usually identify this point with a number or letter. A point has no length, width, or height – it just specifies an exact location. It is zero-dimensional.*

*Every point needs a name. To name a point, we can use a single capital letter. The following is a diagram of points A, B, and M:*

*Lines*

*We can use a line to connect two points on a sheet of paper. A line is one-dimensional. That is, a line has length, but no width or height. In geometry, a line is perfectly straight and extends forever in both directions. A line is uniquely determined by two points.*

*Lines need names just like points do, so that we can refer to them easily. To name a line, pick any two points on the line.*

*The line passing through the points A and B*

*is denoted by *

*A set of points that lie on the same line are said to be collinear.*

*Pairs of lines can form intersecting lines, parallel lines, perpendicular lines and skew lines.*

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*Line segments*

*Because the length of any line is infinite, we sometimes use parts of a line. A line segment connects two endpoints. A line segment with two endpoints A and B is denoted by *

*A line segment can also be drawn as part of a line.*

*Midpoint*

*The midpoint of a segment divides the segment into two segments of equal length. The diagram below shows the midpoint M of the line segment AB. Since M is the midpoint, we know that the lengths AM = MB.*

*Rays*

*A ray is part of a line that extends without end in one direction. It starts from one endpoint and extends forever in one direction.*

*A ray starting from point A and passing through B*

*is denoted by
*

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*Planes*

*Planes are two-dimensional. A plane has length and width, but no height, and extends infinitely on all sides. Planes are thought of as flat surfaces, like a tabletop. A plane is made up of an infinite amount of lines. Two-dimensional figures are called plane figures.*

*All the points and lines that lie on the same plane are said to be coplanar.*

*A plane*

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*Space*

*Space is the set of all points in the three dimensions – length, width and height. It is made up of an infinite number of planes. Figures in space are called solids.*

*Figures in space*

*How to measure angles and types of angles*

*An angle consists of two rays with a common endpoint. The two rays are called the sides of the angle and the common endpoint is the vertex of the angle.*

*Each angle has a measure generated by the rotation about the vertex. The measure is determined by the rotation of the terminal side about the initial side. A counterclockwise rotation generates a positive angle measure. A clockwise rotation generates a negative angle measure. The units used to measure an angle are either in degrees or radians.*

*Angles can be classified base upon the measure: acute angle, right angle, obtuse angle, and straight angle.*

*If the sum of measures of two positive angles is 90°, the angles are called complementary.*

*If the sum of measures of two positive angles is 180°, the angles are called*

**supplementary**.*Examples:*

*1) Two angles are complementary. One angle measures 5x degrees and the other angle measures 4x degrees. What is the measure of each angle?*

*2) Two angles are supplementary. One angle measures 7x degrees and the other measures (5x + 36) degrees. What is the measure of each angle?*

*Straight line intersecting two parallel lines:*

*We will examine now the important aspects of a straight line intersecting two parallel lines.*

*Angles are of three types:*

**Acute angle** – which is less than 900

**Obtuse angle** – which is more than 900

**Right angle** – which is 900

*In any triangle, an external angle is equal to the sum of two of its opposite angles.*

*In a triangle ABC when BC is extended upto D , the exterior angle so formed *

*∠ABD = α Sum of opposite interior angles = β + γ*

*By,*

*Middle School*