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.

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