Congruent angles

Α and β are said to be congruent angles if they measure the same. These have a wide application in the similarity and congruence of different geometric figures; We can classify several of the latter based on their congruent angles, as is the case with some triangles.

Definition

Let α and β be two anglesthese will be congruent angles if they have exactly the same measurement, that is, α = β.

Examples of congruent angles

Vertical angles

When two lines intersect 4 angles are formed. Those that are opposite each other are vertical angles, these are always congruent. Therefore, in the following figure α = β and θ = φ.

Figure 2. Parametric equations of the curve

Alternate angles

A line that cuts two parallels forms congruent angles. In the following figure the pairs ∡ay ∡d; ∡by ∡c; ∡1 and ∡3; ∡4 and ∡2 are congruent.

Alternate Angles Theorem

Isosceles triangle

In this type of triangle two of its internal angles are congruent.

Congruent angles

α = β

Equilateral triangle

In this type of triangle, its three internal angles are congruent and always measure 60⁰.

Congruent angles

α = β = γ

Rectangle and Square

In them, its four internal angles are congruent and measure 90⁰.

Congruent angles Congruent angles4

Application of congruent angles

We can determine the congruence of triangles by using their angles together with the congruence of their sides. Two triangles will be the same if any of the following conditions are met:

  1. If all three sides of two triangles A and B are equal, then they are congruent.
  2. If in two triangles A and B two of their sides and the angle formed by these are equal, then A and B are congruent.
  3. If in two triangles A and B two of their angles are congruent angles and the side between them is equal, then A and B are congruent.
Example:

If the following two triangles are congruent, find the value of a, b, and cꞋ.

Congruent angles

As both are congruent, its three sides must be equal, therefore:

a = aꞋ = 26 cm

b = bꞋ = 20 cm

cꞋ = c = 32 cm