In geometry, antipodal angle is the positional relationship between two angles. The intersection of two straight lines will produce an intersection point, and taking this intersection point as the vertex will produce four angles. Two nonadjacent angles are called antipodal angles. In other words, one of the angles is the inverse angle of the other.

The vertex angle satisfies the following theorem: two straight lines intersect and the vertex angle is equal.

Described in mathematical language is:

Let the straight line AD and BC intersect at point O, then four angles are formed: ∠AOB, ∠COD, ∠AOC, ∠BOD. Where ∠AOB and ∠COD are diagonal, ∠AOC and ∠BOD are diagonal. ∠AOB = ∠COD，∠AOC = ∠BOD .

Properties of antipodal angle

If two angles are diagonal, then they are equal.

On the same plane, two opposite angles are equal.

Examples of vertex angles

As shown in figure 1, two straight lines intersect to form two pairs of vertex angles. ∠ 1 and∠ 3 are a pair of vertex angles, and∠ 2 and∠ 4 are a pair of vertex angles.

note:

1, antipodal angle must be equal, but equal angle is not necessarily antipodal angle.

2. Vertices must have the same vertex.

3. Vertices appear in pairs.

When using the' nature' of vertex angle in the process of proof, take figure 1 as an example.

∴∠ 1=∠3, ∠2=∠4 (equal to the vertex angle).

Clever calculation of vertex angle

Any two straight lines can be regarded as a combination, such a combination has C(n, 2)=n(n- 1)/2, and each combination has two pairs of vertex angles, so n straight lines intersect at one point, and * * * has 2C(n, 2)=n(n- 1) pairs. Namely:

Two straight lines intersect at one point, and there are (2) pairs of different diagonals;

Three straight lines intersect at one point, and there are (6) pairs of different diagonals;

Four straight lines intersect at one point and have (12) pairs of different antipodal angles;

..............

N straight lines intersect at one point, and there are n(n- 1) pairs of different antipodal angles.