Triangle similarity: Definition, criteria and demonstrations

Definition

Two triangles are similar if and only if they have the three angles congruently arranged. Also, the equivalent sides must be proportional.

Similar triangles

Analysis 1

Change the positions of points A, B, C, A', B' or C'. What do you see regarding the value of k?

Analysis 2

Change the positions of points A, B, C, A ', B' or C 'so that k is equal to 1. What can you notice about the triangles?

1st Similarity Case: ANGLE-ANGLE SIMILARITY

2nd Similarity Case: Side-Angle-Side

Exercise

Find a reason for the use of theorem in the previous demonstration: "If a line divides two sides of a triangle and determines segments proportional to those two sides, then it is parallel to the third side." Use the following figure as reference.

3rd Similarity Case: Side-Side-Side (SSS)