1. Introduction to Lines and Angles

INTRODUCTION 
1.    Point: 
      A fine dot made by a sharp pencil or a geometrical figure having no length, breadth and height as called a point.

2.    Line segment
      If we join two fixed points then the figure formed is called line segment.

3.    Line
      If we extend the two end points of a line segment in either direction endlessly then it is called a line.
      It has no definite length.

4.    Ray
      If we extend one end point of a line segment endlessly then it is called a ray.

5.    Angle
      An angle is formed by intersecting two rays. The intersection point is the common initial point of these two rays which is called as vertex of the angle. We use the symbol ‘ ’ to denote the measure of the angle
    •    An angle is an inclination between two rays with the same initial point.

 

 

 

 

 

In this fig. the angles are  

°°

°

°

°

°°

1. Introduction to Lines and Angles

Chapter 5

Lines and Angles

Introduction to Lines and Angles

Line:
A line is a straight figure which doesn’t have an endpoint and extends infinitely in opposite directions.

Ray:
A ray is a straight line, which starts from a fixed point and moves in one direction.

Line Segment
A portion of the line formed with two definite points is called a Line Segment. A line is a one-dimensional figure and has no thickness.

Angle
When we join two line segments at a single point, an angle is formed, or we can say, an Angle is a combination of two line segments at a common endpoint. This common point is called Vertex of the angle and the two line segments are sides or arms of the angle formed.
 

Types of Angles

There are basically 6 types of angles which are:

  1. Acute Angle: If an angle is less than 90 degrees, then it is called an Acute angle
  2. Obtuse Angle: If an angle is more than 90 degrees, then it is called Obtuse Angle
  3. Right Angle: If an angle is exactly at 90 degrees, then it is called Right Angle.
  4. Straight Angle: If an angle is exactly 180 degrees, then it is called Straight Angle.
  5. Reflex Angle: If the angle is more than 180 degree but less than 270 degrees, it is denoted as a Reflex angle.
  6. Full Angle: A 360-degree angle is called a Full angle.

 

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2. Angles on a straight line, intersecting lines

OTHER TYPES OF ANGLES
1.    Complementary angles: If the sum of the measures of two angles is 90° then the angles are complementary angles and each angle is complement of the other angle.

2.    Supplementary angles: If the sum of the measures of two angles is 1800 then the angles are called supplementary angles and each angle is said to be the supplement of the other.

 

Illustration 1 
        Find the measure of an angle which is complement of itself.
 Solution    
        Let the measure of the angle be xº. Then, the measure of its complement is given to be xº.
        Since, the sum of the measures of an angle and its complement is 90º.
           xº + xº = 90º    
           2xº = 90º        
            xº = 45
        Hence, the measure of the angle is 45º.

Illustration 2
        Two supplementary angles differ by 34º. Find the angles.
 Solution
        Let one angle be xº. Then, the other angle is (x + 34)º.
        Now, xº and (x + 34)º are supplementary angles.    
            xº + (x + 34)º = 180º
            2xº + 34º = 180º
            2xº = 180º – 34º
            2xº = 146º
            xº = 73º.
        Hence, the measures of two angles are 73º and 73º + 34º = 107º.

3.    Adjacent angle :
    If a pair of angles have :
    a.    a common vertex
    b.    a common arm
    c.    and their non-common arms are on either side of the common arm. Such pairs of angles are called adjacent angles. Adjacent angles have no common interior points.

4.    Linear pair: A pair of adjacent angles whose non-common sides are opposite rays. The angles in a linear pair are supplementary.

Here OA is common arm and OB and OC form opposite rays.     180°
By seeing this fig. we can say that if a ray stand on a line, then the sum of the two adjacent angles so formed is 180°.

5.    Vertically opposite angles 

 

2. Angles on a straight line, intersecting lines

Angles on a straight line, intersecting lines

Complementary Angle: The sum of the measures of two angles is 90°
Supplementary Angle: The sum of the measures of two angles is 180°
Adjacent Angle: Adjacent angles have a common vertex and a common arm but no common interior points
Linear pair: A linear pair is a pair of adjacent angles whose non-common sides are opposite rays
Vertically Opposite Angles: when two lines intersect, the vertically opposite angles so formed are equal 
Pairs of Lines
Intersecting lines: Two lines intersect if they have a point in common. This common point O is their point of intersection.

Adjacent Angles
A pair of angles which are placed next to each other in such a way that they have a common vertex and an arm which is common to both the angles are called adjacent angles.

Linear Pair
A linear pair is a pair of adjacent angles whose non-common arms are opposite rays.
A pair of angles placed adjacent to each other is called a linear pair if the sum of their measures is 180°.

Note: A pair of supplementary angles form a linear pair when placed adjacent to each other.
When two lines intersect, the vertically opposite angles so formed are equal.

1 + 2 = 180° …(1)
1 and 2 form a linear pair
3 + 2 = 180° …(2)
3 and 2 form a linear pair

From (1) and (2)
1 = 3
Similarly, we can prove that
2 = 4

Intersecting Lines
Two lines intersect if they have a point in common called their point of intersection.

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3. Transversal lines

Pair of lines
1.    Intersecting lines : Two lines are called intersecting lines if they have a common point. This common point is called the point of intersection.

AB and PQ is called the intersecting lines and O is called a point of intersection.

2.    Transversal : A line that intersects two or more lines at distinct points is called a transversal.

In this fig. t is the transversal which cut two lines p and q at distinct points, ‘N’ and ‘M’ respectively.

3.    Perpendicular lines: 

Angles made by a transversal
    Two lines p and q cut by a transversal t. It form angles that have some special names as :

(i)    Exterior angles : The angles whose arms do not contain the segment on the transversal cut by the given lines are called exterior angles. Such as 1, 2 and 7 and 8

(ii)    Interior angles : The angles whose one of the arms contains the segment on the transversal cut by the given lines are called interior angles. Such as 3, 4, 5 and 6

(iii)    Corresponding angles : A pair of angles in which one arm of both the angles of the pair is on the same side of the transversal and the other arms are directed in the same sense is called a pair of corresponding angles. Such as 1 and 5, 4 and                8, 2 and6 and 3 and 7 are corresponding angles.

(iv)    Alternate interior angles : A pair of angles in which one arm of each of the angles is on opposite sides of the transversal and whose other arm contains the segment on the transversal cut by the given lines is called a pair of alternate interior angles.

(v)    Alternate exterior angles : A pair of angles in which one arm of each of the angles is on opposite sides of the transversal and whose other arm is directed in opposite sense doesnot contains the segment on the transversal cut by the given lines is                 called a pair of alternate exterior angles.

3. Transversal lines

Transversal lines

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

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

Transversal of Parallel Lines
The lines on a plane that do not meet anywhere are called parallel lines.
Transversal of parallel lines give rise to quite interesting results.

Angles made by the transversal: There are different angles formed when the transversal cuts the lines. They are:

  • Interior angles
  • Exterior angles
  • Pairs of Alternate interior angles
  • Pairs of Alternate exterior angles
  • Pairs of Corresponding angles
  • Pairs of interior angles on the same side of the transversal

Transversal of Parallel Lines: We know that the parallel lines are the lines that do not meet anywhere. Transversals of parallel lines give rise to quite interesting results.

Checking for Parallel Lines
If a transversal cuts two lines, such that, each pair of corresponding angles are equal in measure.

Similarly, if a transversal cuts two lines, then each pair of the alternate interior angles are equal.
Also, if the transversal cuts the lines, then each pair of interior angles on the same side of the transversal are supplementary.

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Checking for Parallel Lines

Transveral of parallel lines
    Parallel lines are those lines on a plane that do not intersect anywhere.

If two parallel lines are cut by a transversal, then
    •    Each pair of corresponding angles are equal
    •    Each pair of alternate interior angles are equal
    •    Each pair of interior angles on the same side of the transversal are supplementary.

•    If two lines are cut by a transversal and it is given that the pair of corresponding angles are equal or the pair of alternate interior angles or the pair of interior angles are supplementary then the lines have to be parallel.

Illustration 2

Determine the value of x

 

4. Parallel 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.

 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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