Question

prove that sum of all angle of triangle is equal to 180​

Answer 1

Answer:

TRIANGLE

TRIANGLEProof that the sum of the angles in a triangle is 180 degrees

TRIANGLEProof that the sum of the angles in a triangle is 180 degreesTheorem

TRIANGLEProof that the sum of the angles in a triangle is 180 degreesTheoremIf ABC is a triangle then <)ABC + <)BCA + <)CAB = 180 degrees.

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

TRIANGLEProof that the sum of the angles in a triangle is 180 degreesTheoremIf ABC is a triangle then <)ABC + <)BCA + <)CAB = 180 degrees.ProofDraw 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'.

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

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.

OR

Lemma

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

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

LemmaIf ABCD is a quadrilateral and <)CAB = <)DCA then AB and DC are parallel.ProofAssume to the contrary that AB and DC are not parallel.

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

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

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

LemmaIf ABCD is a quadrilateral and <)CAB = <)DCA then AB and DC are parallel.ProofAssume 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 sidesNotice that <)AEC is greater than 0.

LemmaIf ABCD is a quadrilateral and <)CAB = <)DCA then AB and DC are parallel.ProofAssume 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 sidesNotice that <)AEC is greater than 0.Since <)CAB = <)DCA, <)CAE + <)ACE = 180 degrees.

LemmaIf ABCD is a quadrilateral and <)CAB = <)DCA then AB and DC are parallel.ProofAssume 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 sidesNotice that <)AEC is greater than 0.Since <)CAB = <)DCA, <)CAE + <)ACE = 180 degrees.Hence <)AEC + <)CAE + <)ACE is greater than 180 degrees.

LemmaIf ABCD is a quadrilateral and <)CAB = <)DCA then AB and DC are parallel.ProofAssume 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 sidesNotice 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.

$\huge \red{follow \: me}$

Answer 2

Step-by-step explanation:

Draw a line parallel to side BC of the triangle that passes through the vertex A. Label the line PQ. Construct this line parallel to the bottom of the triangle.[1]

Write the equation angle PAB + angle BAC + angle CAQ = 180 degrees. Remember, all of the angles that comprise a straight line must be equal to 180°. Because angle PAB, angle BAC, and angle CAQ combine together to make line PQ, their angles must sum to 180°. Call this Equation 1

State that angle PAB = angle ABC and angle CAQ = angle ACB. Because you constructed line PQ parallel to side BC of the triangle, the alternate interior angles (PAB and ABC) made by the transversal line (line AB) are congruent. Similarly, the alternate interior angles (CAQ and ACB) made by the transversal line AC are also congruent.[3]

Equation 2: angle PAB = angle ABC

Equation 3: angle CAQ = angle ACB

It is a geometric theorem that alternate interior angles of parallel lines are congruent.

Substitute angle PAB and angle CAQ in Equation 1 for angle ABC and angle ACB (as found in Equation 2 and Equation 3) respectively. Knowing that the alternate interior angles are equal lets you substitute the angles of the triangle for the angles of the line.[5]

Thus we get, Angle ABC + angle BAC + angle ACB = 180°.

In other words, in the triangle ABC, angle B + angle A + angle C = 180°. Thus, the sum of all the angles of a triangle is 180°.

HOPE IT WORKS