CBSE Class 10 Mathematics • Chapter 6
Triangles Class 10 Worksheet with Solutions and MCQs
Practise triangle similarity, the Basic Proportionality Theorem and corresponding-side ratios. Use the theorem summary, worked examples, MCQs and FAQs below to improve your understanding and write clear geometry solutions.
This Triangles Class 10 worksheet helps you apply geometry concepts to calculations and proofs. The chapter requires you to recognise equal angles, match corresponding vertices and choose a theorem that fits the given information. A correctly written ratio is often the key to finding a missing length.
Use the worksheet available on this page alongside these explanations and original practice questions. This worksheet on Triangles Class 10 supports homework, classroom revision and independent practice. Before checking an answer, label the given information and write the reason for each important step.
Similar Triangles: Meaning and Correspondence
Two triangles are similar when their corresponding angles are equal and their corresponding sides are proportional. They have the same shape, although their sizes may differ.
Read the Similarity Statement Carefully
If △ABC ∼ △PQR, the vertex correspondence is:
A ↔ P
B ↔ Q
C ↔ R
Therefore:
∠A = ∠P, ∠B = ∠Q and ∠C = ∠R
AB/PQ = BC/QR = AC/PR
Keep this order when writing ratios. Do not match sides only because they appear horizontal or vertical in a drawing.
Triangles Class 10 Theorems and Similarity Criteria
| Theorem or Criterion | What You Need to Show |
|---|---|
| AA similarity | Two angles of one triangle are respectively equal to two angles of another triangle. |
| SSS similarity | All three pairs of corresponding sides are proportional. |
| SAS similarity | Two pairs of corresponding sides are proportional and their included angles are equal. |
| Basic Proportionality Theorem, or BPT | A line parallel to one side of a triangle divides the other two sides in the same ratio. |
| Converse of BPT | A line dividing two sides of a triangle in the same ratio is parallel to the third side. |
Triangles Class 10 Formulas and Useful Ratios
BPT Configuration
In △ABC, let D lie on AB and E lie on AC, with DE ∥ BC. Then:
AD/DB = AE/EC
Also, △ADE ∼ △ABC by AA similarity, giving:
AD/AB = AE/AC = DE/BC
Remember that AB = AD + DB and AC = AE + EC. A ratio involving a whole side is different from a ratio involving only the remaining segment.
For similar triangles, corresponding perimeters have the same ratio as corresponding sides. Their areas have the square of that ratio. For example, if the side ratio is 2:3, the area ratio is 4:9. Use additional results according to the scope of your assigned textbook.
Class 10 Triangles Important Questions with Solutions
Example 1: Find a Missing Length Using BPT
Question: In △ABC, D lies on AB and E lies on AC. If DE ∥ BC, AD = 4 cm, DB = 6 cm and AE = 5 cm, find EC.
By the Basic Proportionality Theorem:
AD/DB = AE/EC
4/6 = 5/EC
4 × EC = 30
EC = 7.5 cm
Check:
4/6 = 2/3
5/7.5 = 2/3
Example 2: Apply the Converse of BPT
Question: D and E lie on sides AB and AC of △ABC. Given AD = 3 cm, DB = 5 cm, AE = 6 cm and EC = 10 cm, determine whether DE is parallel to BC.
AD/DB = 3/5
AE/EC = 6/10
AE/EC = 3/5
Since AD/DB = AE/EC, the two sides are divided in the same ratio. Therefore, DE ∥ BC by the converse of BPT.
Example 3: Find a Corresponding Side
Question: If △ABC ∼ △PQR, AB = 6 cm, PQ = 9 cm and BC = 8 cm, find QR.
Corresponding sides of similar triangles are proportional:
AB/PQ = BC/QR
6/9 = 8/QR
6 × QR = 72
QR = 12 cm
The scale factor from △ABC to △PQR is 9/6 = 3/2. Multiplying BC by 3/2 also gives 12 cm.
Example 4: Prove Similarity Using AA
Question: In triangles ABC and PQR, ∠A = 50°, ∠B = 60°, ∠P = 50° and ∠Q = 60°. Prove that the triangles are similar.
∠A = ∠P = 50°
∠B = ∠Q = 60°
Two corresponding angles are equal. Therefore, △ABC ∼ △PQR by AA similarity.
Their third angles are also equal because the angles of each triangle add to 180°.
Triangles Class 10 Case Study: Height from Shadows
A vertical pole 2 metres high casts a shadow 1.5 metres long. At the same time, a nearby vertical tree casts a shadow 9 metres long on level ground. Find the height of the tree.
The pole and tree form right triangles with their shadows. Assuming the sunlight has the same direction at both locations, the angles made by the sun’s rays with the ground are equal. Hence, the triangles are similar by AA.
Let the tree’s height be h metres.
h/2 = 9/1.5
h/2 = 6
h = 12 metres
The shadow-length scale factor is 6, so the corresponding height is also multiplied by 6.
Triangles Class 10 MCQ with Answers
1. Which criterion uses two equal corresponding angles?
A. SSS B. AA C. SAS D. RHS congruence
Answer: B. AA. Two equal corresponding angles are sufficient to establish triangle similarity.
2. Are all equilateral triangles similar?
A. Yes B. No C. Only if their sides are equal
Answer: A. Yes. Every angle in an equilateral triangle is 60°, so any two equilateral triangles are similar by AA.
3. If △ABC ∼ △PQR, which ratio is correct?
A. AB/QR = BC/PQ
B. AB/PQ = BC/QR
C. AB/PR = BC/PQ
Answer: B. AB corresponds to PQ, and BC corresponds to QR.
4. Do proportional sides 3, 4, 5 and 6, 8, 10 establish similarity?
A. Yes, by SSS B. No C. Only by AA
Answer: A. 3/6 = 4/8 = 5/10 = 1/2, so the triangles are similar by SSS.
Extra Class 10 Triangles Questions for Practice
- In △ABC, DE ∥ BC, with D on AB and E on AC. If AD = 5 cm, DB = 10 cm and AE = 4 cm, find EC.
- If △ABC ∼ △PQR, AB = 8 cm, PQ = 12 cm and AC = 10 cm, find PR.
- Two triangles have corresponding side lengths 5, 7, 9 and 10, 14, 18. State the similarity criterion.
- Two triangles have angle measures 40°, 65°, 75° and 65°, 75°, 40°. Are they similar? Give a reason.
- Explain why two triangles with equal corresponding angles need not be congruent.
Check the Practice Answers
- EC = 8 cm, using 5/10 = 4/EC.
- PR = 15 cm.
- SSS similarity; every corresponding-side ratio is 1/2.
- Yes, by AA similarity after matching equal angles.
- Their shapes match, but their sizes may differ. Congruence requires equal corresponding side lengths.
How to Revise Triangles Class 10 Exercises and PYQs
Begin with the meaning of similarity and vertex correspondence for Triangles Class 10 Exercise 6.1. Practise BPT and its converse for Exercise 6.2, then focus on similarity criteria and related proofs for Exercise 6.3. Follow the exercise numbering and content in your own NCERT edition.
When using Triangles Class 10 NCERT solutions 6.2 or 6.3, attempt each question before reading the solution. Check the theorem used, the order of corresponding vertices and the calculation. Apply the same approach to NCERT Exemplar questions.
A Triangles Class 10 previous year questions PDF or a test paper can help you practise under timed conditions. Choose papers with a clearly identified board and year. The examples and extra questions above are original practice questions.
Common Mistakes in Triangle Solutions
- Writing a similarity statement in the wrong vertex order.
- Mixing corresponding and non-corresponding sides in a ratio.
- Using BPT without establishing that the relevant lines are parallel.
- Using SAS without checking the included angle.
- Assuming two triangles are similar because they look alike.
- Giving a proof without stating reasons for angle or ratio equalities.
FAQs on Triangles Class 10 Worksheet
What are similar triangles?
Similar triangles have equal corresponding angles and proportional corresponding sides. For example, triangles with side lengths 3, 4, 5 and 6, 8, 10 are similar because each side of the second is twice its corresponding side in the first.
What is the difference between similarity and congruence?
Similar triangles have the same shape. Congruent triangles have the same shape and size. Every pair of congruent triangles is similar, but similar triangles are congruent only when their corresponding side ratio is 1.
What is the Basic Proportionality Theorem?
If a line parallel to one side of a triangle intersects the other two sides, it divides those sides in the same ratio. For D on AB and E on AC, DE ∥ BC gives AD/DB = AE/EC. This is commonly labelled Theorem 6.1 in NCERT Chapter 6.
When should I use the converse of BPT?
Use the converse when you know that two sides are divided in the same ratio and need to prove the joining line is parallel to the third side. For example, AD/DB = AE/EC gives DE ∥ BC in the standard triangle configuration.
What are the similarity criteria for Class 10 triangles?
The main criteria are AA, SSS and SAS. AA uses two equal corresponding angles; SSS uses three proportional corresponding-side pairs; SAS uses two proportional side pairs and an equal included angle.
Why are two equal angles enough to prove similarity?
Each triangle’s angles add to 180°. If two corresponding angles are equal, the third angles must also be equal. The triangles are therefore similar by AA.
Are all isosceles triangles similar?
No. Isosceles triangles can have different vertex angles. For example, one can have angles 40°, 70°, 70°, while another has 80°, 50°, 50°. Their corresponding angles do not match. All equilateral triangles, however, are similar.
Is SSA a general triangle similarity criterion?
No. Two proportional side pairs and an equal non-included angle do not establish similarity in every case. For SAS, the equal angle must lie between the two sides being compared.
How do I avoid confusing AD/DB with AD/AB?
DB is the remaining segment, while AB is the whole side. In the BPT configuration, AD/DB = AE/EC. Similarity of △ADE and △ABC gives AD/AB = AE/AC. Keep segment-to-segment and segment-to-whole comparisons consistent.
Which Triangles Class 10 important questions should I practise?
Practise BPT calculations, proving parallelism with its converse, AA/SSS/SAS proofs, missing corresponding sides and applications such as height from shadows. Include questions where you must explain why a theorem applies.
Are Similar Triangles Exercises 8.1, 8.2 and 8.4 the same as NCERT Chapter 6?
Exercise numbering differs between boards and textbooks. CBSE NCERT uses Chapter 6 for Triangles. If your book uses Similar Triangles Exercises 8.1, 8.2 or 8.4, match solutions to that book’s exact questions rather than assuming the numbering refers to NCERT.
Where can I find Triangles Class 10 notes and a mind map?
Visit the Triangles Class 10 Notes and Mind Map on WitKnowLearn. Review the theorem statements, then return to this worksheet to practise their application.
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