Triangles Class 10
Download the PDF containing Triangles Class 10 notes and a mind map for chapter revision. Understand similar triangles, the Basic Proportionality Theorem and similarity criteria through clear explanations, worked examples and student FAQs.
Two triangles can have the same shape even when one is larger than the other. This idea of similarity helps you calculate unknown lengths, compare figures and solve practical problems involving shadows or scale drawings. In Class 10 Triangles, the important skill is recognising the correct relationship and explaining why it applies.
These Triangles Class 10 notes support your 2026 revision with theorem statements, a formula table and step-by-step examples. Read the explanations first, use the Triangles Class 10 mind map for recall, and then practise questions independently. Pay particular attention to the order of corresponding vertices when writing a similarity statement.
Similar Triangles and Corresponding Parts
Two triangles are similar when their corresponding angles are equal and their corresponding sides are proportional. If △ABC ∼ △PQR, the order tells you that A corresponds to P, B to Q and C to R.
Read the Similarity Statement Carefully
∠A = ∠P, ∠B = ∠Q, ∠C = ∠R
AB/PQ = BC/QR = AC/PR
Match sides using their endpoints. For example, AB corresponds to PQ because A matches P and B matches Q. The apparent position or orientation of the triangles does not determine correspondence.
Similarity and congruence are different. Congruent triangles have the same shape and size. Similar triangles have the same shape, while their size may differ. Congruent triangles are therefore similar with a scale factor of 1.
Triangles Class 10 Similarity Criteria
| Criterion | Required Information | Key Check |
|---|---|---|
| AA similarity | Two corresponding angles are equal. | The third angles are automatically equal. |
| SSS similarity | All three corresponding side ratios are equal. | Use matching sides consistently. |
| SAS similarity | Two corresponding side pairs are proportional and their included angles are equal. | The equal angle must lie between the selected sides. |
Example 1: Check Similarity by SSS
Question: One triangle has sides 4 cm, 6 cm and 8 cm. Another has sides 6 cm, 9 cm and 12 cm. Are they similar?
4/6 = 2/3
6/9 = 2/3
8/12 = 2/3
All corresponding side ratios are equal.
Therefore, the triangles are similar by SSS.
Basic Proportionality Theorem Class 10
The Basic Proportionality Theorem, or BPT, states that a line parallel to one side of a triangle, intersecting the other two sides at distinct points, divides those sides in the same ratio.
In △ABC, let D lie on AB and E lie on AC. If DE ∥ BC, then: AD/DB = AE/EC. Notice that this compares the two parts of each side.
Example 2: Find an Unknown Length Using BPT
Question: In △ABC, D lies on AB and E lies on AC, with DE ∥ BC. If AD = 3 cm, DB = 5 cm and AE = 4.5 cm, find EC.
By BPT, AD/DB = AE/EC.
3/5 = 4.5/EC
3EC = 22.5
EC = 7.5 cm
BPT Proof Using Areas
For the same configuration, join BE and CD. Let ar(△) denote the area of a triangle. Use the fact that triangle areas with a common altitude are proportional to their bases.
Step-by-Step Proof
△ADE and △BDE have bases AD and DB on the same line,
and the same altitude from E.
ar(△ADE)/ar(△BDE) = AD/DB
△ADE and △CDE have bases AE and EC on the same line,
and the same altitude from D.
ar(△ADE)/ar(△CDE) = AE/EC
△BDE and △CDE share base DE and lie between the parallel
lines DE and BC.
Therefore, ar(△BDE) = ar(△CDE).
Hence, AD/DB = AE/EC.
Converse of the Basic Proportionality Theorem
The converse reverses the direction of the result. If a line intersects two sides of a triangle and divides them in the same ratio, it is parallel to the third side.
Example 3: Prove That Two Lines Are Parallel
Question: D and E lie on AB and AC respectively. If AD = 4 cm, DB = 6 cm, AE = 6 cm and EC = 9 cm, prove DE ∥ BC.
AD/DB = 4/6 = 2/3
AE/EC = 6/9 = 2/3
Therefore, AD/DB = AE/EC.
By the converse of BPT, DE ∥ BC.
Triangles Class 10 Formulas and Useful Relationships
| Relationship | Result | Condition |
|---|---|---|
| Corresponding sides | AB/PQ = BC/QR = AC/PR | △ABC ∼ △PQR |
| BPT | AD/DB = AE/EC | D on AB, E on AC and DE ∥ BC |
| Smaller triangle to whole triangle | AD/AB = AE/AC = DE/BC | △ADE ∼ △ABC |
| Perimeter ratio | Ratio of perimeters = corresponding side ratio | Similar triangles |
| Area ratio | ar(△ABC)/ar(△PQR) = (AB/PQ)2 | Similar triangles; useful for related geometry practice |
Do Not Mix Part-to-Part and Part-to-Whole Ratios
AD/DB compares the two parts of AB, while AD/AB compares a part with the whole side. Write matching relationships on both sides of an equation. If AD = 3 and DB = 5, then AB = 8, so these ratios are 3/5 and 3/8 respectively.
Triangles Class 10 Questions with Solutions
Example 4: Find a Corresponding Side
Question: △ABC ∼ △PQR. If AB = 6 cm, PQ = 9 cm and BC = 8 cm, find QR.
AB/PQ = BC/QR
6/9 = 8/QR
6QR = 72
QR = 12 cm
Example 5: Find a Height Using Shadows
Question: A vertical 1.5 m pole casts a 2 m shadow. At the same time, a vertical tree casts a 12 m shadow on level ground. Find the tree’s height.
Both height–shadow triangles have a right angle and the same
angle made by the sun’s rays.
They are similar by AA.
Tree height/12 = 1.5/2
Tree height = 12 × 1.5/2
Tree height = 9 m
How to Write a Similarity Proof
- Identify the two triangles and the result you need to establish.
- Mark the given equal angles, parallel lines and side lengths.
- Write the angle equalities or side ratios with reasons.
- State the similarity criterion: AA, SSS or SAS.
- Write the triangles in the correct corresponding order.
- Use the required side proportion to complete the proof.
Never assume triangles are similar because a drawing looks convincing. Use given information, angle properties or a proved relationship. A short proof with clear reasons is more useful than a long list of unexplained equations.
Triangles Class 10 Mind Map and Revision 2026
Use the mind map of Triangles Class 10 to recall similarity criteria, BPT, its converse and corresponding-side ratios. Cover one branch and explain the theorem aloud, including its conditions. Then solve a matching numerical or proof question.
For 2026 revision, use the textbook and syllabus applicable to your course. Match question wording when practising Triangles Class 10 Exercise 6.1, 6.2 or 6.3 because editions can differ. Follow routine exercises with suitable NCERT Exemplar questions and verified previous year questions.
Quick Practice Questions and MCQs
-
Similar triangles have corresponding side ratio 3:5.
Find their perimeter ratio.
Answer: 3:5. -
Similar triangles have corresponding side ratio 2:3.
Find their area ratio.
Answer: 4:9. -
In △ABC, DE ∥ BC. If AD = 2 cm, DB = 3 cm and AE = 4 cm,
find EC.
Answer: 6 cm.
Triangles Class 10 MCQ with Answers
1. Two pairs of corresponding angles are equal.
Which similarity criterion applies?
A. SSS B. AA C. SAS D. None
Answer: B.
2. All congruent triangles are:
A. Similar B. Equilateral C. Right-angled D. Isosceles
Answer: A. Their corresponding side ratio is 1.
Practise with Related Worksheets and Notes
Apply the theorems using the Triangles Class 10 Worksheet with Solutions and MCQs . Attempt each question before checking its solution.
Explore more revision resources in the CBSE Class 10 Maths Notes and Mind Maps collection .
Frequently Asked Questions
Does this Triangles Class 10 PDF contain notes and a mind map?
Yes. The PDF combines notes for understanding the chapter with a mind map for visual revision. Use the notes first, then recall theorem conditions and relationships through the map.
What is the difference between similar and congruent triangles?
Similar triangles have the same shape, with equal corresponding angles and proportional corresponding sides. Congruent triangles also have the same size. Similarity does not require a side ratio of 1.
Is AA the same as AAA similarity?
Both establish similarity. Two equal corresponding angles are sufficient because the angles in each triangle total 180°, making the third pair equal automatically.
How do I identify corresponding sides?
Use the equal-angle pairs and the vertex order in the similarity statement. If △ABC ∼ △PQR, then AB matches PQ, BC matches QR and AC matches PR. Do not match sides only by their position on the page.
What is the difference between BPT and its converse?
BPT starts with a parallel line and concludes that two sides are divided proportionally. Its converse starts with proportional division and concludes that the joining line is parallel to the third side.
Can I prove BPT using similar triangles?
That argument can be valid when similarity has been established independently. In the NCERT development, similarity results follow BPT, so using those results to prove BPT can be circular. The area-based proof above avoids that issue.
How do I know which similarity criterion to use?
Use AA when two angle pairs are equal, SSS when three side ratios are equal, and SAS when two side ratios and their included angle match. Start by listing exactly what is given or can be proved.
Are two proportional sides enough to prove similarity?
No. For SAS similarity, the angles between those sides must also be equal. Two proportional side pairs and an unrelated equal angle are not generally sufficient.
Are all equilateral triangles similar?
Yes. Every equilateral triangle has three 60° angles, so any two are similar by AA. They are congruent only when their side lengths are equal.
Does equal area mean that two triangles are similar?
No. Equal area alone does not determine shape. You still need an appropriate similarity criterion. Likewise, equal perimeter by itself does not establish similarity.
Why is the area ratio the square of the side ratio?
In similar triangles, corresponding bases and corresponding heights scale by the same factor k. Since area is half the product of base and height, the area scales by k².
How can I improve at Triangles Class 10 proof questions?
Redraw and label the figure, list useful angle relationships and practise writing correspondence correctly. Include a reason for each equality and name the similarity criterion. Review mistakes in logic as carefully as arithmetic mistakes.
Where can I practise Triangles Class 10 important questions with solutions?
Use the Triangles Class 10 worksheet with solutions and MCQs after revising these notes. Combine numerical questions with proof practice to strengthen both calculation and reasoning.
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