prove that sum of all the sides of a triangle is 180
Answers
Answer:
Angles B and B' are corresponding angles, ... So, the angle sum ∠A + ∠B + ∠C is equal to the angle sum ∠A' + ∠B' + ∠C'. The three angles A', B', and C' form together a straight angle (they are along the line l). So, their angle sum is 180°.
Step-by-step explanation:
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.
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:
ANSWER IN CLIPPING
Step-by-step explanation:
PLEASE MARK ME AS BRILLIANCE PLEASE
