Theorem : The sum of the angles of a triangle is 180°

-

Given              A △ABC
To Prove         ∠1 + ∠2 + ∠3 = 180°
Construction  Through A, draw a line DE parallel to BC.
Proof               DE ॥ BC and AB is the transversal.
                        ∴    ∠1 = ∠4   (Alt. Int. angle)   ............ (i)
                        Again, DE ॥ BC  and AC is the transversal.
                        ∴    ∠2 = ∠5   (Alt. Int. angle)   ............ (ii)
                        On adding the corresponding sides of  (i) and (ii)
                        ∴    ∠1 + ∠2  = ∠4 + ∠5   
                        On adding ∠3 on both sides
                        So, ∠1 + ∠2 + ∠3 = ∠4 + ∠5 + ∠3
                        But, ∠4 + ∠5 + ∠3 = ∠ DAE = 180° 
                                                            ( ∵  ∠DAE is a straight angle)
                        ∴    ∠1 + ∠2 + ∠3 = 180°
                        Hence, the sum of the angles of a triangle is  180° . 

Comments

Post a Comment

Popular posts from this blog

What is Decimal Numbers? Terminating and Non-terminating | Recurring and Non-recurring decimal

-
Image
Decimal Number Decimal numbers are numbers that are written according to their place value. The decimal point is used to separate whole part from number value less than one.A symbol is used " . ", this is called decimal point. Example: 65.894 where 65 is a whole part and 894 is a fractional part. Type of Decimals 1. Terminating Decimals  2. Non-terminating Decimals 1. Terminating Decimals Every fraction a/b can be expressed as a decimal. If the decimal expression of a/b terminates i.e. comes to end, then the decimal so obtained is called a terminating decimal. Example: 1/2 = 0.5                   5/4 = 1.25                 13/5= 2.6 Thus, each of the numbers 1/2, 5/4 and 13/5 can be expressed in the form of a terminating decimal. 2. Non-terminating Decimals While expressing a fraction in the form of decimal, when we perform division we get some remainder. If the division process does not end i.e. remainder is not equal to zero, then such dec
Read more

What is angle | Type of angles

-
Image
Angle   Two rays starting from a common point form an angle. In ∠ AOB, O is the vertex, OA  and OB  are the two rays . − → . Measure of a angle   The  amount of turning from OA to OB is called the measure of ∠ AOB , written as m∠ AOB.  An angle is measured is degrees denoted by ‘°’. Types of angles (i)                   Acute angle: An angle whose measure is greater than 0° but less than 90° is called an acute angle . Example: 15° ,30° ,60° ,75° ,etc.   ∠ AOB  = 30° is an acute angle.                       (ii)                 Right angle: An angle whose measure is 90° is called a right angle .          ∠AOB  = 90°is a right angle. (iii)                Obtuse angle: An angle whose measure is greater than 90° and less than 180° is called an obtuse angle . Example: 100°, 110°, 120°, 140°, etc.   ∠AOB  = 110° is an obtuse angle.              (iv)               Straight angle: When the rays of an angle are opposite rays forming a stra
Read more