Question

The sum of angles of a triangle is 180. prove​

Answer 1

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.

Definition

Two Triangles ABC and A'B'C' are congruent if and only if

|AB| = |A'B'|, |AC| = |A'C'|, |BC| = |B'C'| and,

<)ABC = <)A'B'C', <)BCA = <)B'C'A', <)CAB = <)C'A'B'.

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Answer 2

Answer:

Here's a slightly different, but common, approach that might help you (or someone else!) out.

Let the triangle be ABC.

Extend the line AB beyond B to D.

Draw the line BE parallel to AC, with C and E on the same side of AB.

See the diagram attached.

Then:

∠DBE = ∠CAB   since BE is parallel to AC (the "F" rule).

∠CBE = ∠BCA   since BE is parallel to AC (the "Z" rule).

The sum of the angles is then

∠ABC + ∠BCA + ∠CAB

= ∠ABC + ∠CBE + ∠DBE

= 180°    since the angles form a straight line.