Introduction: Lines & Angles | Quantitative Reasoning for GRE PDF Download

Introduction 

What is a line?

A line is a single-dimensional or one-dimensional figure, with no thickness, and extends in both directions infinitely. Usually, it is called a straight line just to emphasize that there are no curves anywhere in it.

In the above picture, AB and BC are two lines meeting at point B. The point B is called the Vertex. The space between the two lines marked as “b” is called the Angle.
Other Definitions included as part of the lines are listed below for immediate reference. 

• Line segment: A part of a line with two endpoints is called a line segment. Line segment AB is denoted by AB. 
• Ray: A part of a line with one endpoint is called a ray. 
• Collinear points and non-collinear points: If three or more than three points are on the same line, then they are called collinear points, otherwise, they are non-collinear points. 
• Intersecting Lines and Non-intersecting Lines: Lines PQ and RS are intersecting lines because they are intersecting with each other at O.  Lines AB and CD are non-intersecting (parallel) lines.

Note: The lengths of the common perpendicular at different points on these parallel lines are the same. This equal length is called the distance between two parallel lines.

What are Angles?

An angle is a space or the gap formed between two lines that meet at a point and the meeting point is called the Vertex. The two lines originating from the same point are called the arms of the angle.
The Angles are classified into different types such as Acute angle, Right angle, Obtuse angle, Straight angle, Reflex angle, and Full angle. These angles are explained in the table given below along with images showing different angles.
When a pair of angles are formed, they are named differently based on a few criteria. The names of the pair of angles and their criteria are given below.

• Complementary Angles: Two angles whose sum is 90° are called complementary angles. 
• Supplementary Angles: Two angles whose sum is 180° are called supplementary angles. 
• Adjacent Angles: Two angles are adjacent if they have a common vertex, a common arm and their non-common arms are on different sides of the common arm. ∠ABD and ∠DBC are the adjacent angles. Ray BD is their common arm and point B is their common vertex. Ray BA and Ray BC are non-common arms.
  Note: ∠ABC = ∠ABD + ∠DBC 
• Vertically Opposite Angles: The vertically opposite angles formed when two lines intersect each other at a point. Two lines AB and CD intersect each other at point O, then, there are two pairs of vertically opposite angles. One pair is ∠AOD and ∠BOC and another pair is ∠AOC and ∠BOD.
  If two lines intersect each other, then the vertically opposite angles are equal.  ∠AOD = ∠BOC; ∠COA = ∠DOB
• Linear Pair of Angles: When the sum of two adjacent angles is 180°, then they are -
  called a linear pair of angles.
  (i) If a ray stands on a line, then the sum of two adjacent angles so formed is 180°.
  (ii) If the sum of two adjacent angles is 180°, then a ray stands on a line (that is the non-common arms form a line). In the above image, Ray OC stands on line AB. ∠AOC + ∠BOC = 180°