Chapter 6: Triangles
Similarity and Its Conditions
This chapter introduces the concept of similar figures—shapes that have the same form but not necessarily the same size. All congruent figures are similar, but not all similar figures are congruent. For polygons, two figures are similar if i) their corresponding angles are equal and ii) the lengths of their corresponding sides are in the same ratio. The concept is visualized using real-life objects like photographs and shadows. The scale factor (or ratio) plays a key role in identifying and comparing similar figures.
Similarity of Triangles and Theorems
For triangles, several theorems and activities help establish when two triangles are similar. The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. Its converse is also true: if a line divides two sides in the same ratio, it must be parallel to the third side. These theorems are foundational in proving triangle similarity and solving geometric problems.
Criteria for Triangle Similarity
The chapter outlines three main criteria for triangle similarity:
- AAA (or AA): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
- SSS: If all corresponding sides of two triangles are in the same ratio, the triangles are similar.
- SAS: If one angle is equal and the sides including the angle are in proportion, the triangles are similar.
These criteria help in solving problems involving proportions, parallel lines, heights, and shadows, often connecting geometry to real-life scenarios.