Question

Prove that the sum of the angle a triangle is 180.

Answer 1

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.

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.

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'.

Answer 2

Answer:

hope it will helps you mate.