Chapter Hub
Triangles
This chapter focuses on how triangles compare in shape, how parallel lines divide sides, and how similarity helps in proving many useful relations. For Class 10 CBSE, the main ideas are similarity criteria, the Basic Proportionality Theorem, area ratio, and right-triangle relations. You should learn the exact conditions first, then practice applying them in numerical problems and proof-based questions. A good revision of this chapter means knowing when to use angle comparison, side ratios, square checks, and scale factor ideas correctly.
Difficulty
Medium
Study time
96-120 min
Plan by time
Pick the window that matches what you have right now.
If you have 15 min
Last-pass revision
Skim the Quick Revision table — definitions, formulas, and the traps board examiners reuse.
Open Quick RevisionIf you have 45 min
Targeted practice
Read the high-priority concepts, then take the chapter MCQ quiz to find weak spots.
Start MCQ QuizIf you have 96 min
First full pass
Walk every concept in chapter order, then revise and quiz. Best for the first time you study this chapter.
Open Key ConceptsChapter Learning Map
Start with one of the buckets below, then open the full map when you want the complete concept roadmap.
Similarity and criteria
Covers Similarity of triangles, AAA similarity criterion, SAS similarity criterion, SSS similarity criterion and Scale factor in similar triangles.
Open mapParallel line results
Covers Basic proportionality theorem, Converse of basic proportionality theorem, Using similarity to prove relations and Area ratio of similar triangles.
Open mapRight triangle relations
Covers Pythagoras theorem, Converse of Pythagoras theorem and Altitude to hypotenuse relations.
Open mapKey Concepts
Concepts grouped the way the chapter is taught — open the group that matches what you want to revise.
Similarity and criteria
Covers Similarity of triangles, AAA similarity criterion, SAS similarity criterion, SSS similarity criterion and Scale factor in similar triangles.
Similarity of triangles
Two triangles are similar when their corresponding angles are equal and their corresponding sides are in the same ratio.
AAA similarity criterion
If three angles of one triangle are respectively equal to three angles of another triangle, the triangles are similar.
SAS similarity criterion
Two triangles are similar if one included angle is equal and the two sides around that angle are proportional.
SSS similarity criterion
Two triangles are similar if all three corresponding sides are proportional.
Scale factor in similar triangles
The scale factor is the number by which one similar triangle is enlarged or reduced to get the other.
Parallel line results
Covers Basic proportionality theorem, Converse of basic proportionality theorem, Using similarity to prove relations and Area ratio of similar triangles.
Basic proportionality theorem
A line drawn parallel to one side of a triangle divides the other two sides in the same ratio.
Converse of basic proportionality theorem
If a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.
Using similarity to prove relations
Once two triangles are proved similar, you can use corresponding side ratios and equal angles to prove lengths, proportions, and parallel-line results.
Area ratio of similar triangles
For similar triangles, the ratio of their areas equals the square of the ratio of their corresponding sides.
Right triangle relations
Covers Pythagoras theorem, Converse of Pythagoras theorem and Altitude to hypotenuse relations.
Pythagoras theorem
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Converse of Pythagoras theorem
If the square of the longest side of a triangle equals the sum of the squares of the other two sides, the triangle is right-angled.
Altitude to hypotenuse relations
In a right triangle, the altitude from the right angle to the hypotenuse creates two smaller triangles that are similar to the original triangle and to each other.
Exam Intelligence
The traps that cost marks in this chapter, and how its main concepts turn up in the paper.
High Probability Topics
- Similarity of triangles
- AAA similarity criterion
- SAS similarity criterion
- SSS similarity criterion
- Basic proportionality theorem
- Converse of basic proportionality theorem
- Area ratio of similar triangles
- Pythagoras theorem
Common Traps
- Mixing up corresponding vertices and writing ratios in the wrong order.
- Using side equality where only proportionality is needed.
- Forgetting to square the side ratio for area ratio questions.
- Using Pythagoras theorem in a triangle that is not right-angled.
- Checking the wrong side in the converse of Pythagoras theorem.
- Adding hypotenuse segments instead of using the product relation in altitude problems.
How CBSE Asks This Chapter
- Similarity of triangles
CBSE questions usually ask whether two triangles are similar from angle data, side ratios, or a short proof-based figure.
- AAA similarity criterion
A proof or short-answer question may give angles and ask for the correct similarity reason.
- SAS similarity criterion
A figure may give one equal angle and two side lengths around it, then ask whether the triangles are similar.
- SSS similarity criterion
A question may give only side lengths and ask whether similarity can be proved without angle information.
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
Remember
Corresponding angles are equal, and corresponding sides are proportional in similar triangles.Trap to avoid
Students often think similar triangles must have equal side lengths.
Remember
AAA similarity: three corresponding angles equal implies similarity.Trap to avoid
Students sometimes wait for side lengths even after all the corresponding angles have already matched.
Remember
SAS similarity: one included angle equal and the two surrounding sides proportional.Trap to avoid
Students often compare the wrong pair of sides or forget that the angle must lie between the compared sides.
Remember
SSS similarity: all three corresponding sides are proportional.Trap to avoid
Students sometimes check only two side ratios or mix up the side order.
Remember
If DE ∥ BC in triangle ABC, then AD/DB = AE/EC.Trap to avoid
Students often write the side ratio in the wrong order or forget that the line must be parallel.
Converse of basic proportionality theorem
Remember
If AD/DB = AE/EC, then DE ∥ BC.Trap to avoid
Students think the line is parallel only after measuring angles.
Area ratio of similar triangles
Remember
Area ratio of similar triangles = (corresponding side ratio)2.Trap to avoid
Students often keep the same ratio for area instead of squaring it.
Remember
c2 = a2 + b2, where c is the hypotenuse in a right triangle.Trap to avoid
Students may apply the formula to a triangle that is not right-angled, or choose the wrong side as the hypotenuse.
Converse of Pythagoras theorem
Remember
If c is the longest side and c2 = a2 + b2, then the triangle is right-angled.Trap to avoid
Students sometimes check the wrong side or forget that the largest side must be treated as the hypotenuse.
Altitude to hypotenuse relations
Remember
If the hypotenuse is split into p and q, then h2 = pq; each leg also satisfies leg2 = hypotenuse × adjacent segment.Trap to avoid
Students often add the two hypotenuse segments or use the wrong side relation.
Using similarity to prove relations
Remember
If triangles are similar, then corresponding sides are proportional and corresponding angles are equal.Trap to avoid
Students often stop after writing the similarity statement and do not use the corresponding side ratios.
Scale factor in similar triangles
Remember
Side ratio = scale factor; perimeter ratio = scale factor; area ratio = (scale factor)2.Trap to avoid
Students sometimes reverse the factor or use it directly for area without squaring.
| Concept | Remember | Trap to avoid |
|---|---|---|
| Similarity of triangles | Corresponding angles are equal, and corresponding sides are proportional in similar triangles. | Students often think similar triangles must have equal side lengths. |
| AAA similarity criterion | AAA similarity: three corresponding angles equal implies similarity. | Students sometimes wait for side lengths even after all the corresponding angles have already matched. |
| SAS similarity criterion | SAS similarity: one included angle equal and the two surrounding sides proportional. | Students often compare the wrong pair of sides or forget that the angle must lie between the compared sides. |
| SSS similarity criterion | SSS similarity: all three corresponding sides are proportional. | Students sometimes check only two side ratios or mix up the side order. |
| Basic proportionality theorem | If DE ∥ BC in triangle ABC, then AD/DB = AE/EC. | Students often write the side ratio in the wrong order or forget that the line must be parallel. |
| Converse of basic proportionality theorem | If AD/DB = AE/EC, then DE ∥ BC. | Students think the line is parallel only after measuring angles. |
| Area ratio of similar triangles | Area ratio of similar triangles = (corresponding side ratio)2. | Students often keep the same ratio for area instead of squaring it. |
| Pythagoras theorem | c2 = a2 + b2, where c is the hypotenuse in a right triangle. | Students may apply the formula to a triangle that is not right-angled, or choose the wrong side as the hypotenuse. |
| Converse of Pythagoras theorem | If c is the longest side and c2 = a2 + b2, then the triangle is right-angled. | Students sometimes check the wrong side or forget that the largest side must be treated as the hypotenuse. |
| Altitude to hypotenuse relations | If the hypotenuse is split into p and q, then h2 = pq; each leg also satisfies leg2 = hypotenuse × adjacent segment. | Students often add the two hypotenuse segments or use the wrong side relation. |
| Using similarity to prove relations | If triangles are similar, then corresponding sides are proportional and corresponding angles are equal. | Students often stop after writing the similarity statement and do not use the corresponding side ratios. |
| Scale factor in similar triangles | Side ratio = scale factor; perimeter ratio = scale factor; area ratio = (scale factor)2. | Students sometimes reverse the factor or use it directly for area without squaring. |
Practice
Use short concept checks first, then move into the full chapter test.
Free Chapter MCQ Quiz
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