Lines and Angles

Introduction to Lines and Angles

Basic Terms and Definitions


Line

A line is a collection of points along a straight path. A line extends in both the directions and has no endpoints. It has no definite length.

Line segmentA line segment is a part of a line with two end points. There is only one line segment joining two given points.

Ray
A part of line with one end point is called a ray.

Collinear points All the points that lie on the same line are called collinear points.

Angle
Two rays originating from the same end point form an angle.
Consider the two rays AB and AC originating from the same point A form an angle named as ∠BAC or ∠CAB or ∠A. The rays that form the angle are called the arms of the angle. The end point is called the vertex of the angle. The size of an angle is measured in degrees.

Types of angles 


Acute angle
Angles measuring less than 90° but more than 0° are called acute angles.
Right angle
An angle measuring 90° is called a right angle. A right angle is formed by the intersection of two  perpendicular lines.
Obtuse angle
Angles measuring greater than 90° but less than 180° are called obtuse angles.

Straight angle
An angle measuring 180° is called a straight angle.

Reflex angle
Angles measuring greater than 180° but less than 360° are called reflex angles.
Adjacent angles
 Two angles are said to be adjacent if they have a common arm and a common vertex.

Linear pair of angles
Two adjacent angles are said to form a linear pair if their sum is 180°. Non-common arms of the linear pair of angles form a straight line

Vertically opposite angles
When two lines intersect, four angles are formed. The angles that are opposite to each other are called vertically opposite angles. There are two pairs of vertically opposite angles.

Complementary angles
Two angles are said to be complementary, if their sum is 90°.

Supplementary angles Two angles are said to be supplementary if their sum is 180°.

Complementary and supplementary angles may or may not be adjacent angles.

Intersecting linesTwo or more lines that meet at one point are called intersecting lines. When two lines intersect each other, then the vertically opposite angles are equal. Sum of all the angles formed at a point is 360°.

Parallel lines
Lines on the same plane that never intersect are called parallel lines.

Parallel Lines

Intersecting lines
Two lines having only one point in common are called intersecting lines


Parallel lines
Two lines drawn in a plane are parallel if they do not intersect even when they are produced. Distance between the parallel lines is the same.



Transversal
A line that intersects two or more lines, at different points is called a transversal.

Introduction to Lines and Angles, Parallel Lines, NCERT Revision Notes Class 9 (IX) Mathematics, NCERT solution, NCERT Solved Question Answers.

When a transversal intersects two lines, m and n, eight angles are formed, four angles at each point, P and Q respectively. These angles are identified by their positions.


        •  ∠1, ∠2, ∠7 and ∠8 are called exterior angles
        •  ∠3, ∠4, ∠5 and ∠6 are called interior angles
        •  ∠1 and ∠5, ∠2 and ∠6, ∠4 and ∠8, ∠3 and ∠7 are pairs of corresponding angles
        •  ∠1 and ∠7,  ∠2 and ∠8 are pairs of alternate exterior angles
        •  ∠4 and ∠6, ∠3 and ∠5 are pairs of alternate interior angles
        •  ∠4 and ∠5, ∠3 and ∠6 are consecutive interior angles on the same side of the transversal

If a transversal intersects two parallel lines, then

        •  each pair of corresponding angles is equal.
        •  each pair of alternate interior angles is equal.
        •  each pair of interior angles on the same side of the transversal is supplementary



Two lines intersected by a transversal are parallel if, either
        •  any one pair of corresponding angles is equal, or
        •  any one pair of alternate interior angles is equal, or
        •  any one pair of interior angles on the same side of the transversal is supplementary



Lines which are parallel to the same line are parallel to each other.



The sum of three angles of a triangle is 180°.



If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.


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