Triangles
CBSE Class 10 · Mathematics · Notes, formulas and practice questions
Revision page on similarity of triangles for CBSE Class 10 Mathematics: similar figures, Basic Proportionality Theorem (Thales) with converse, AAA/AA, SSS and SAS similarity criteria, and problems using ratios of corresponding sides.
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Similar figures have exactly the same shape but not necessarily the same size. For triangles, similarity is determined by three equal corresponding angles and equal ratios of corresponding sides. This chapter builds tools to prove two triangles are similar without checking every angle and side: the Basic Proportionality Theorem (also called Thales' theorem) gives a way to detect parallel lines inside a triangle, and the AAA/AA, SSS and SAS criteria give shortcuts for similarity. Throughout, careful attention to which angles and sides correspond to each other is essential.
The Basic Proportionality Theorem states that when a line is drawn parallel to one side of a triangle and it intersects the other two sides, it divides those two sides in the same ratio. Its converse is equally important: if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side. This converse is often used to prove that two line segments are parallel. In numerical problems, setting up the correct proportion is key — for example, writing AD/DB = AE/EC when DE is parallel to BC, or using the alternate form AD/AB = AE/AC when complete side lengths are involved.
Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. The AAA criterion says equal corresponding angles alone are enough; in triangles, AA is sufficient because the remaining angle follows from the angle-sum property. The SSS criterion checks that all three pairs of corresponding sides have the same ratio. The SAS criterion checks two pairs of sides in proportion and the included angle between them equal. Once similarity is established, all ratios of corresponding sides, and even ratios of perimeters, follow automatically.
In examinations, similarity problems usually appear as length-finding or parallel-line-proving questions. The most common student errors are using the wrong pair of corresponding sides, swapping numerators and denominators, and failing to check that the angle in SAS is between the proportional sides. Always begin by writing the similarity statement with matching vertices in order, such as ΔABC ∼ ΔPQR. This small step fixes the correspondence and turns a confusing word problem into a straightforward proportion that can be solved by cross-multiplication.
Key terms
- Similar figures
- Figures having exactly the same shape but not necessarily the same size. In similar polygons, all corresponding angles are equal and all corresponding side lengths are in the same ratio.
- Similar triangles
- Two triangles for which corresponding angles are equal and all three pairs of corresponding sides are proportional. The triangles are usually written with matching vertices in order, as ΔABC ∼ ΔPQR.
- Basic Proportionality Theorem (Thales' theorem)
- In a triangle, a line drawn parallel to one side divides the other two sides proportionally. If D is on AB and E is on AC of ΔABC, and DE ∥ BC, then AD/DB = AE/EC.
- Converse of Basic Proportionality Theorem
- If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. It is used to prove that segments are parallel.
- AAA similarity criterion
- Two triangles are similar if the three angles of one are respectively equal to the three angles of the other. Angle equality forces all corresponding side ratios to be equal.
- AA similarity criterion
- Two triangles are similar when any two corresponding angles are equal, because the third angles then automatically become equal by the angle-sum property of a triangle.
- SSS similarity criterion
- Two triangles are similar if the three sides of one are proportional to the three corresponding sides of the other, meaning all three side ratios are equal.
- SAS similarity criterion
- Two triangles are similar when two sides of one are proportional to two corresponding sides of the other, and the included angles between those sides are equal.
Formula sheet
| What | Formula | Notes |
|---|---|---|
| Side ratio for similar triangles | AB/DE = BC/EF = CA/FD | For ΔABC ∼ ΔDEF. Corresponding sides are paired by the order of vertices in the similarity statement. This equality can be used to find an unknown side. |
| Basic Proportionality Theorem | AD/DB = AE/EC | In ΔABC, with D on AB and E on AC, DE ∥ BC. An equivalent form is AD/AB = AE/AC. |
| Converse of BPT | AD/DB = AE/EC ⇒ DE ∥ BC | Here D and E are on AB and AC of ΔABC. The equality of the two ratios is the condition that forces DE to be parallel to BC. |
Practice questions with answers
1. In ΔABC, D is on AB and E is on AC. DE ∥ BC, AD = 4 cm, DB = 6 cm, AE = 3.2 cm. Find EC.
From the Basic Proportionality Theorem, AD/DB = AE/EC, so 4/6 = 3.2/EC. Cross-multiplying gives EC = 6 × 3.2 / 4 = 4.8 cm.
2. In ΔPQR, M and N are on sides PQ and PR such that PM/MQ = PN/NR = 3/7. Is MN parallel to QR? Give reason.
Yes. The converse of the Basic Proportionality Theorem states that if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side. Since MN divides PQ and PR in the same ratio, MN ∥ QR.
3. State the definition of similarity for two triangles.
Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio.
4. ΔABC has sides AB = 3 cm, BC = 4 cm, CA = 5 cm, and ΔDEF has sides DE = 4.5 cm, EF = 6 cm, FD = 7.5 cm. Prove that the triangles are similar.
Take the side ratios in corresponding order: AB/DE = 3/4.5 = 2/3, BC/EF = 4/6 = 2/3, and CA/FD = 5/7.5 = 2/3. All three ratios are equal, so by the SSS similarity criterion, ΔABC ∼ ΔDEF with correspondence A↔D, B↔E, C↔F.
5. In two similar triangles, the corresponding side lengths are in the ratio 4 : 9. What is the ratio of their perimeters?
Since every side of the first triangle is 4/9 of the corresponding side of the second, the perimeter of the first is also 4/9 of the perimeter of the second. Hence the ratio of their perimeters is 4 : 9.
6. A vertical stick 1.2 m long casts a shadow 3 m long. At the same time, a building casts a shadow 18 m long. Find the height of the building.
Sunlight strikes both objects at the same angle, so the right triangles formed by height and shadow are similar. Thus 1.2/3 = h/18, giving h = 1.2 × 18 / 3 = 7.2 m. The building is 7.2 m tall.
7. In ΔXYZ, ∠X = 50° and ∠Y = 70°. In ΔPQR, ∠P = 50° and ∠R = 60°. Are the two triangles similar? Explain.
Yes. The third angle of ΔXYZ is ∠Z = 180° − 50° − 70° = 60°, and the third angle of ΔPQR is ∠Q = 180° − 50° − 60° = 70°. Thus ∠X = ∠P, ∠Z = ∠R, and the remaining angles are equal too, so the triangles are similar by the AA criterion.
8. In ΔABC, D lies on AB with AD : DB = 2 : 3, and E lies on AC with AE : EC = 2 : 3. Prove that DE ∥ BC.
The ratios AD/DB and AE/EC are both 2/3, so the line DE divides the two sides AB and AC in the same ratio. By the converse of the Basic Proportionality Theorem, DE must be parallel to BC.
9. If ΔABC ∼ ΔDEF, ∠D = 60° and ∠F = 80°, and AB = 2 × DE, find ∠B.
In similar triangles corresponding angles match, so ∠A = ∠D = 60° and ∠C = ∠F = 80°. Therefore ∠B = 180° − 60° − 80° = 40°. The side ratio AB/DE = 2 does not affect the angles.
10. The sides of one triangle are 14 cm, 18 cm and 24 cm. The shortest side of a similar triangle is 7 cm. Find the remaining two sides of the second triangle.
The ratio of the shortest sides is 14 : 7 = 2 : 1, so the second triangle is half the size of the first. Its other sides are 18/2 = 9 cm and 24/2 = 12 cm.
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