Answer:
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
IABI = IA'B', IACI = |A'C'I, IBCI = IB'C'| and, <)ABC = <)A'B'C', <)BCA = <)B'C'A', <)CAB =
<)C'A'B'.
Hope this helps!
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