Triangles and Similarity Lecture Notes

Key vocabulary terms and geometric theorems regarding triangle similarity and related properties based on the provided text.

Congruent figures

Two figures that have the same shape and the same size.

Similar figures

Two figures having the same shape but not necessarily the same size.

Scale factor

The same ratio of the corresponding sides of two similar polygons, also referred to as the Representative Fraction.

Equiangular triangles

Triangles where the corresponding angles of the two triangles are equal.

Thales Theorem (Basic Proportionality Theorem)

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

AAA (Angle-Angle-Angle) criterion

A similarity criterion stating that if in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the triangles are similar.

AA similarity criterion

A specific case of similarity stating that if two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

SSS (Side-Side-Side) similarity criterion

A criterion stating that if in two triangles, sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the triangles are similar.

SAS (Side-Angle-Side) similarity criterion

A criterion stating that if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the triangles are similar.

RHS Similarity Criterion

A criterion stating that if in two right triangles, the hypotenuse and one side of one triangle are proportional to the hypotenuse and one side of the other triangle, then the two triangles are similar.

Similarity of Polygons

Two polygons of the same number of sides are similar if: (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio or proportion.

Theorem 6.2

The converse of the Basic Proportionality Theorem, stating that if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.