Chapter 6: Triangles
Two triangles are similar if they have the same shape but possibly different sizes. The cleaner definition: corresponding angles are equal and corresponding sides are in the same ratio. From this tiny seed grows an enormous theory , the basic proportionality theorem (often called Thales' theorem), three concise similarity criteria, a beautiful result about the ratio of areas of similar triangles, and ultimately a slick proof of the Pythagoras theorem and its converse.
For board exams this chapter is geometry-heavy and proof-rich. Expect -mark MCQs on identifying similar triangles, - to -mark applications of the basic proportionality theorem (BPT), - to -mark proofs (similarity criteria, area ratio, Pythagoras), and a couple of construction-flavoured questions. Diagrams matter , clean, labelled figures earn marks.
In real life, similar triangles power every scale drawing, every map, every architectural plan, every photograph. Shadows, mirrors, the very way the eye works , all rest on this idea. Pythagoras shows up in carpentry, surveying, navigation, and computer graphics.
The chapter is heavy on careful argument. Every step should be justified by a numbered reason: a hypothesis, a definition, a known theorem, or an algebraic manipulation. We will build the habit through the worked examples.
What's inside
- Similar polygons and triangles , definitions and basic examples.
- Basic Proportionality Theorem (Thales) , and its converse.
- Similarity criteria , AAA / AA, SSS, SAS for triangles.
- Areas of similar triangles , ratio of areas ratio of squares of corresponding sides.
- Pythagoras theorem , statement, proof using similarity, and the converse.
Key results / Formula card
| Result | Statement |
|---|---|
| BPT (Thales) | A line parallel to one side of a triangle divides the other two sides in the same ratio. |
| Converse of BPT | If a line divides two sides of a triangle in the same ratio, it is parallel to the third side. |
| AAA / AA | Two triangles with equal corresponding angles are similar (knowing two pairs equal is enough). |
| SSS | Two triangles whose corresponding sides are proportional are similar. |
| SAS | Two triangles with one pair of equal angles and the including sides proportional are similar. |
| Area ratio | for similar triangles. |
| Pythagoras | In a right triangle, . |
| Converse of Pythagoras | If for sides of a triangle, the angle opposite is a right angle. |