There are 8 ways to prove the sum of triangle angles and triangle angles

Through the triangle ABC, a vertex A makes a straight line AD intersects the BC edge at the point D, and then passes the vertices B and C to make a straight BE and BF parallel to AD respectively

According to the parallel of two straight lines, the inner misalignment angle is equal:

Angular bad=angular abe;

Angular cad = angular acf;

Because be cf;

Angle EBC + angle FCB = 180 degrees;

And because the angle eba + angle abc = angle ebc;

Angle FCA + Angle ACB = FCB Angle;

So the angle eba + angle abc + angle fca + angle acb = 180 degrees;

And because the angle bad + angle cad = angle abe + angle acf = angle bac;

So the angle abc + angle acb + angle bac = 180 degrees;

So there is: the sum of the inner angles of the triangle is equal to 180 degrees.

Suppose the triangle is ABC

Extend the BC and mark a D point.

Because the outer angle ACD of angle c is equal to the sum of the two angles that are not adjacent to it.

So acd=a+b

Because ACD+C=180 (the sum of an angle and its outer angle is a flat angle, i.e., 180), A+B+C=180

Proof: Passing the point a as mn||bc

So the angle mab = angle b (two straight lines are parallel and the inner wrong angles are equal) so the angle nac = angle c (two straight lines are parallel and the inner wrong angles are equal) because mn is a straight line (made).

So angle mab + angle nac + angle bac = 180 degrees.

Therefore, the angle b + angle c + angle bac = 180 degrees (equal substitution), so the sum of the internal angles of the triangle is equal to 180 degrees.

Today's midterm exam, just finished, this is the process of our teacher to write, there should be no big problem. Those symbols really can't be typed.,It's good to change it yourself when you prove it.,Also,"Because""So "to align with":",I don't know what's going on.,I know that it's not aligned = =+.

Hope it helps

Butterflies have claws.

To prove that the sum of the inner angles of the triangle is 180°, in addition to measuring the degrees of the angles with a protractor and then adding them together, there are other dynamic methods to prove it.

Prove the inner angle of the triangle and 180° by folding and splicing, the method is simple, direct and easy to understand!Students are more receptive.

Methods to prove triangles:It is not a "mathematical proof" that can be verified by the method of measurement and splicing, and because there are countless triangles with different shapes, it is impossible for us to verify all the triangles one by one by the above method, so it is convincing to verify it by making parallel lines!

Application: Two straight lines are parallel and the inner wrong angles are equal.

Flat angle definition. Application: Two straight lines are parallel and the inner wrong angles are equal.

The two straight lines are parallel and the isotope angles are equal.

Flat angle definition. From the parallel:1+ 2+ 4=180° (two straight lines are parallel, complementary to the inner angles) and 3= 4 (two straight lines are parallel, the internal wrong angles are equal).

From this derives the sum theorem of the triangle row inner angle: the triangle inner angle sum 180°.

1.Place three triangles of the same size at the positions of the three corresponding corners, and mark the letters a, b, and c. Then put together the A angle of the first triangle, the B angle of the second triangle, and the rotten high C angle of the third triangle, so that the bottom edge is exactly formed a straight line.

That is, the three corners form a flat angle. Therefore, the sum of the degrees of the angles of the Sanxun family is one hundred and eighty degrees, that is, it proves that the sum of the internal angles.

2.Extend one side of the triangle to form the outer corner of the triangle. This angle and the inner angle of the triangle adjacent to it are added to the flat angle, so they are adjacent complementary angles.

Then make a straight line parallel to the opposite side of the corner at the vertex of this inner corner, dividing the outer corner into two corners. By using two straight lines that are parallel, the isotope angles are equal, and the inner misalignment angles are equal, it can be proved that the other two angles of the triangle are equal to the two angles separated from the outer angles. Then the sum of the three interior angles of the triangle is equal to the one of the inner angles plus its adjacent complementary angles, which is one hundred and eighty degrees.

1.Fold the three corners of a triangle inward, and the three corners just form a flat angle, so it is 180 degrees.

2.Make parallel lines to opposite sides at one vertex and prove with an inner wrong angle.

Do triangle ABC

Crossing point A is a straight line EF parallel to BC

Angle EAB = Angle B

Angle FAC = Angle C

Angular EAB + angular FAC + angular BAC = 180

Angle BAC + Angle B + Angle C = 180

4.The sum of the internal angle and formula (n-2) * 180

5.Let the three vertices of the triangle be a, b, and c, corresponding to the angles a, b, and c, respectively. The passing point A makes a straight line L parallel to the straight line BC, and the angle between L and ray AB is B', L and ray AC form an angle of C', angle b'with angle B, angle C'According to the equality theorem of the inner wrong angles of parallel lines, the sum of the internal angles of the triangle = angle a + angle b + angle c = angle a + angle b'+ Angular C'= 180 degrees.

6.Extend the sides of the triangle ABC, DAB=C+B, EBA=A+C, FCA=A+B

So dab+eba+fca=2a+2b+2c=360 (the sum of the outer angles of the triangle is 360).

So a+b+c=180

7.Extend one side of the triangle shape to form a triangular diplomacy. It is easy to see that this angle and the inner angle of the triangle adjacent to it add up to a flat angle (180 degrees), so they are adjacent complementary angles.

Then raise the apex of the inner corner of the cherry blossom to make a straight line parallel to the opposite side of the corner, and divide that diplomacy into two brigades of jujube corners. By using two parallel lines, equal isotope angles, and equal internal wrong angles, it can be proved that the other two angles of the triangle are equal to the two angles separated by this diploma. Then the sum of the three inner angles of the triangle is equal to the inner angle plus its adjacent complementary angle, that is, 180 degrees.

8.Place three triangles of the same size at the positions of the three corresponding angles and mark them with the letters a, b, and cThen put together the A angle of the first triangle, the B angle of the second triangle, and the C angle of the third triangle, so that the bottom (or top) of them forms a straight line.

That is, the three corners form a flat angle. That is to say, the sum of the degrees of the three angles is one hundred and eighty degrees. And these three angles are the three inner angles of the triangle.

Another method of proving the theorem of the number of circumferential angles.

The theorem of the number of circumferential angles is an important theorem in the chapter of the circle, it is an important basis for solving the problem of angles related to the circle, the proof of this theorem is given in the Beijing edition of the mathematics textbook, this proof method is mainly used in the knowledge of the outer angles, the teachers in the teaching are mostly modeled on the method in the book to prove, and rarely go to ** and think about other proof methods, the following is given with the inner angle of the triangle and the method of proving this theorem, for your reference

Verification: The circumferential angle of the same arc is equal to half of the number of angles to the center of the circle

It is known that o, AOB and ACB are the central and circumferential angles of the pair, respectively

Verification: AOB=2 ACB

oc=ob，oc＝oa

oca=∠oac，∠ocb=∠obc

oca+∠oac+∠aoc=180°，∠ocb+∠obc+∠boc=180°

aoc=180°-∠oca-∠oac，∠boc=180°-∠ocb-∠obc

aoc=180°-2∠oca，∠boc=180°-2∠ocb

aoc-∠boc =180°-2∠oca-180°+2∠ocb

aoc-∠boc =2(∠ocb -∠oca)

AOC- BOC = AOB, OCB - OCA = Socks ACB

aob=2∠acb；

oc=ob，oc＝oa

oca=∠oac，∠ocb=∠obc

oca+∠oac+∠aoc=180°，∠ocb+∠obc+∠boc=180°

aoc=180°-∠oca-∠oac，∠boc=180°-∠ocb-∠obc

aoc=180°-2∠oca，∠boc=180°-2∠ocb

aoc+∠boc+∠aob =360°

aob=360°-∠aoc-∠boc

aob=360°-180°+2∠oca-180°+2∠ocb

aob=2(∠oca+∠ocb)

oca+∠ocb =∠acb

aob=2∠acb ；

To sum up, the circumferential angle of an arc is equal to half of the central angle of the circle it opposes.

Evidence 1: As the extension of BC CD, through the point C as CE BA, then 1= A, 2= B, and 1+ 2+ ACB=180° A+ B+ ACB=180°

Evidence 2: If the point c is de ab, then 1= b, 2= a, 1+ acb+ 2=180° a+ acb+ b=180°

Proof 3: Take a little D on BC, as de ba to ac, df ca to f, then there is 2 = b, 3 = c, 1 = 4, 4 = a. ∴∠1=∠a。

1+ 2+ 3=180° a+ b+ c=180°

Arbitrary n-sided interior angles and formulas.

The sum of the internal angles of any n-sided is =180°· (n-2)。 where is the sum of the inner angles of the n-sided polygon, and n is the number of sides of the polygon. From one vertex of a polygon and the other vertices, the polygon can be divided into (n-2) triangles, and the sum of the internal angles of each triangle is 180°, so the formula for the sum of the internal angles of any n-sided is :

n-2)·180°，∀n=3，4，5，…。

1) Center of gravity: the intersection point of the three midlines, the distance from this point to the vertex is twice the distance from it to the midpoint of the opposite side; The center of gravity to the median ratio is 1:2;

2) Perpendicular: The intersection of the three high lines of the triangle is called the perpendicular center of the triangle.

3) Heart: The intersection of the three bisectors of the inner angles of the triangle is called the heart of the triangle. That is, the center of the inscribed circle, which is at an equal distance to the three sides.

4) Outer center: refers to the intersection point of the perpendicular bisector of the three sides of the triangle, also known as the perpendicular line. is an abbreviation for the center of the circumscribed circle of the triangle, which is at an equal distance from the three vertices.

5) Side center: the intersection point of an inner angle bisector line and the other two outer angle bisector lines (there are three in total), which is the abbreviation of the center of the side tangent circle of the triangle.

To prove that the sum of the inner angles of the triangle is 180°, in addition to measuring the degree of the angle with a protractor, and then adding it, there are other dynamic methods to prove it.

By folding and splicing socks to prove the inner angle of the triangle and 180°, the method is simple, direct and easy to understand! Students are more receptive.

Methods to prove triangles:It is not a "mathematical proof" that is not completely convincing, and since there are countless triangles with different shapes, it is impossible to verify all the triangles one by one with the above method, so it is convincing to verify this lead by making a "mathematical proof" of parallel lines!

Application: Two straight lines are parallel and the inner wrong angles are equal.

Flat angle definition. Application: Two straight lines are parallel and the inner wrong angles are equal.

The two straight lines are parallel and the isotope angles are equal.

Flat angle definition. From the parallel:1+ 2+ 4=180° (two straight lines are parallel, complementary to the inner angles) and 3= 4 (two straight lines are parallel, the internal wrong angles are equal).

From this derives the sum theorem of the triangle row inner angle: the triangle inner angle sum 180°.