Geometry: Points, Lines, and Angles

Fundamental Concepts

• Geometry deals with points, lines, angles, and shapes.
• This chapter focuses on points, lines, angles, and their properties.

Point

• A point is a basic geometric object with position but no length or breadth.
• Points are represented by dots and denoted by capital letters (e.g., A, B, C).

Line

• A straight line (or simply a line) is a collection of points with position, shape, and length, but negligible breadth and thickness.
• A line extends infinitely in both directions.
• Lines are denoted by two points on the line (e.g., AB) or a small letter (e.g., l, m, n).

Ray

• A ray is a part of a straight line that extends infinitely in one direction from a fixed point.
• A ray has one end point, called the initial point.
• Rays are denoted by the initial point and another point on it (e.g., AB for ray AB).

Line Segment

• A line segment is a part of a straight line between two points.
• A line segment has two endpoints and a finite length
• The line segment between points A and B is denoted as AB, and its length is the distance between A and B
• The midpoint of a line segment AB point C where AC = CB = ½ AB

Properties of Lines

• Infinite number of points exist on a straight line.
• Infinite number of straight lines can pass through a single point.
• Only one straight line passes through two different points.
• Two different straight lines can intersect at only one point (or have only one point in common).

Collinear Points

• Three or more points are collinear if they lie on the same straight line.

Concurrent Lines

• Three or more straight lines are concurrent if they have one point in common, called the point of concurrency.

Plane

• A plane is a flat surface with length and breadth but no thickness.
• Extends infinitely
• Made up on an infinite number of lines and points
• If two points on a line lie in a plane, then the entire line lies in the plane.

Angles

• An angle is formed by two rays sharing a common endpoint (vertex).
• The symbol is ∠.
• The angle AOB is denoted ∠AOB or ∠BOA (vertex in the middle).

Unit of Measurement

• Angles are measured by the amount one arm has to turn about the vertex to reach the other arm.
• Degrees (°) are the unit of measurement. A complete turn is 360°.
• 1 degree is divided into 60 minutes (1° = 60'), and 1 minute is divided into 60 seconds (1' = 60").

Classification of Angles

• Right angle: An angle of 90°.
• Acute angle: An angle less than 90°.
• Obtuse angle: An angle greater than 90° but less than 180°.
• Straight angle: An angle of 180°
• Reflex angle: An angle greater than 180° but less than 360°.
• Zero angle: An angle of 0° (arms coincide).

Adjacent Angles

• Adjacent angles have the same vertex, a common arm, and their other arms on different sides of the common arm.

Linear Pair

• The sum of adjacent angles is 180°, and the outer arms are on the same line
• The adjacent angles that add up to 180°

Complementary Angles

• Two angles are complementary if the sum of their magnitudes is 90°.

Supplementary Angles

• Two angles are supplementary if the sum of their magnitudes is 180°.

Angles at a Point

• If a ray meets a straight line at a point on the line, the sum of the magnitudes of the adjacent angles so formed is 180°.
• The sum of all the angles at a point is 360°.

Vertically Opposite Angles

• When two lines intersect, the vertically opposite angles formed are equal.

Bisector of an Angle

• A ray that divides an angle into two equal parts is called the bisector of the angle.

Bisector of a Line Segment

• The line EC, which passes through C (the midpoint of line segment AB) and divides the line segment into two equal parts AC and CB, is called a bisector of AB.