Question

prove that sum of all angles of a triangle is 180 degree

Answer 1

TRIANGLE
Proof that the sum of the angles in a triangle is 180 degrees
Theorem
If ABC is a triangle then <)ABC + <)BCA + <)CAB = 180 degrees.

Proof
Draw line a through points A and B. Draw line b through point C and parallel to line a.

triangle
Since lines a and b are parallel, <)BAC = <)B'CA and <)ABC = <)BCA'.
It is obvious that <)B'CA + <)ACB + <)BCA' = 180 degrees.
Thus <)ABC + <)BCA + <)CAB = 180 degrees.

Lemma
If ABCD is a quadrilateral and <)CAB = <)DCA then AB and DC are parallel.

Proof
Assume to the contrary that AB and DC are not parallel.
Draw a line trough A and B and draw a line trough D and C.
These lines are not parallel so they cross at one point. Call this point E.

four sides
Notice that <)AEC is greater than 0.
Since <)CAB = <)DCA, <)CAE + <)ACE = 180 degrees.
Hence <)AEC + <)CAE + <)ACE is greater than 180 degrees.
Contradiction. This completes the proof.

Answer 2

Hi friend..
Here is your ANSWER




➡️➡️✴️SOLUTION✴️ ⬅️⬅️


To prove this we are given a triangle PQR and Angle 1, Angle 2 and Angle 3 are the angles of triangle PQR.

To prove Angle 1 + A angle 2 + Angle 3 = 180 degree, draw a line XYP parallel to QR through the opposite vertex P.

XPY is a line,

Therefore, Angle 4 + Angle 1 + angle 5 = 180 degree                              (1)

XPY || QR and PQ,PR are transvesals 

Angle 4 = Angle 2   &

Angle 5 = Angle 3 ( Pairs of alternate angles)

Substituting Anggle 4 and Angle 5 in (1) we get

Angle 2 + Angle 1 + Angle 3 = 180 dgree  

i.e. Angle 1 + Angle 2 + Angle 3 = 180 degree



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