7th Class Mathematics Lines and Angles Parallel Lines

     Parallel Lines                

Two lines are said to be parallel if the distance among them always remains same at each and every point. The parallel lines never intersect each other.               

In other words we can say that if two lines do not have any common point than they are said to be parallel. In the figure I and m are parallel lines.                

       Concept of Transversal                

Transversal is a line which intersects two or more parallel lines. In the figure, n is a transversal line.                

      Alternate Interior Angles                

In the above figure                

\[\angle 3\]and \[\angle 5,\text{ }\angle 4\]and \[\angle 6\]are alternate interior angles.  

      Alternate Exterior Angles                                                 

In the above figure \[\angle 7\]and \[\angle 2\]are alternate exterior angles.                  

      Corresponding Angles

\[\angle 2\] and \[\angle 3\] in the above figure are corresponding angles.                

They are also the angles on the same side of transversal.                  

         Properties of Angles                

When the parallel lines are intersected by a transversal:

•    Corresponding angles are equal.
•    Alternate interior angles are equal.
•    The sum of interior angles on the same side of transversal is \[180{}^\circ .\]  

  Using the figure below which one of the following statements is true?                

(a) Z ABD and \[\angle ABC\]are adjacent angles                

(b) \[\angle ABD\]and \[\angle DBC\]are complementary                

(c) \[\angle DBC\]is half of the measure of \[\angle ABC\]                

(d) \[\angle ABC\]and Z DBC are congruent                

(e) None of these                

Answer: (b)                       

Explanation                

From the figure only option (b) is correct because\[\angle ABD+\angle DBC={{90}^{O}}\]                

  Find the value of x in the figure given below.                

(a)\[15{}^\circ \]                                              

(b) \[20{}^\circ \]                

(c)\[9{}^\circ \]                                                

(d) \[15{}^\circ \]                

(e) None of these                                

Answer: (d)                

Explanation                

From the figure \[5x{}^\circ +5x{}^\circ +2x{}^\circ =180{}^\circ \]                

\[\Rightarrow 12x{}^\circ =180{}^\circ \Rightarrow X{}^\circ =15{}^\circ \]                

  Find the difference between two angles in the figure given below.                

(a)\[15{}^\circ \]                                              

(b) \[50{}^\circ \]                

(c)\[70{}^\circ \]                                              

(d) \[20{}^\circ \]                

(e) None of these                

Answer: (b)                

Read the following statements.                

(i) \[\angle PQT\] and \[\angle TQS\] are adjacent angles                

(ii) \[\angle TQR\] and \[\angle RQS\] are adjacent angles                

(iii) \[\angle TQP\] and \[\angle TQR\] are linear pairs               

(iv) \[\angle SQR\] and \[\angle SQT\] are linear pairs                

Which one of the following options represents the incorrect statement?                

(a) (i), (ii)                                            

(b) (ii), (iii)                

(c) (ii), (iv)                                          

(d) (i), (iv)                

(e) None of these                

Answer: (c)  

  The angle formed between the bisectors of linear pair is always:                

(a) An acute angle                

(b) An obtuse angle                

(c) An angle which is half of the double of the right angle                

(d) An acute angle greater than the half of the right angle                

(e) None of these                                

Answer: (c)        

• One and only line can be drawn between two fixed points.
• If the sum of measure of two angles is \[90{}^\circ \] then they are said to be complementary angles.
• If the sum of measure of two angles is \[180{}^\circ \] then they are said to be supplementary angles.
• Two angles are said to be adjacent angles, if                

(i)   They have a common vertex.                

(ii)  They have common arm.                

(iii) Non common arms is opposite to the common arm.

• If the sum of measure of two adjacent angles is \[180{}^\circ \] then they are said to be linear pair of angles. In the linear pair non-common arms are opposite to each other.    

• In the third century BC Euclid was the mathematician who put geometry into an axiomatic form
• Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus.
• The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies served as an important source of geometric problems during the next one and a half millennia.
• A mathematician who works in the field of geometry is called a geometer.