E Board: Suggested Books     ICSE Board:  Foundation Mathematics 
Class 8: Reference Books     Class 8: NTSE Preparation 
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Properties of Triangles:

a. Angle Sum Property

Theorem: The sum of the angles of a triangle is 180°.p1

Proof:

Given: \Delta ABC

To Prove: \angle ABC+\angle BCA+\angle CAB=180^{\circ}

First draw a line DE parallel to BC (DE \parallel BC) .

AB \ and\ AC are the transversals.

Since DE \parallel BC ,

\angle ABC=\angle BAD (Alternate Angles)……………………….i)

Similarly,

\angle BCA=\angle CAE  (Alternate Angles)…………………………ii)

Adding i) and ii), we get

\angle ABC+\angle BCA=\angle BAD+\angle CAE …………………………iii)

Adding \angle BAC on both sides of iii) we get the following

\angle ABC+\angle BCA+\angle BAC=\angle BAD+\angle CAE+\angle BAC

Since \angle BAD+\angle CAE+\angle BAC=180^{\circ}  (Angles on a straight line)

Hence

\angle ABC+\angle BCA+\angle BAC=180^{\circ}

or \angle A+\angle B+\angle C=180^{\circ}

b. Exterior Angle Property

Theorem: If one side of the triangle is produced, then the exterior angle so formed is equal to the sum of the interior opposite angles.p2

Proof:

To Prove: \angle ACD=\angle CAB+\angle BAC

Given: Given: \Delta ABC

Extend BC\ to\ D . Also draw a like CE \parallel AB.\ AC\ \&\  BC are the transversals.

Therefore, \angle ACE=\angle BAC (Alternate Angles)

Similarly,\angle DCE=\angle ABC  (Corresponding Angles)

\angle ACD=\angle ACE+\angle ECD=\angle ABC+\angle BAC

or \angle ACD=\angle ABC+\angle BAC

Hence proved, that if one side of the triangle is produced, then the exterior angle so formed is equal to the sum of the interior opposite angles.

E Board: Suggested Books     ICSE Board:  Foundation Mathematics 
Class 8: Reference Books     Class 8: NTSE Preparation 
--------------------------------------------------------------

 

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