Keerthana Classes

Maths ch6 triangles

October 04, 2026

37 marks. Read the notes, then tap Start test. Answers and your score appear only after you submit.

Quick notes

Figures with the same shape but not necessarily the same size are similar. All congruent figures are similar, but similar figures need not be congruent.

Two triangles are similar if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio.

Basic Proportionality Theorem (Thales): if a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. In ΔABC with DE ∥ BC: AD/DB = AE/EC.

Converse: if a line divides two sides of a triangle in the same ratio, it is parallel to the third side.

Similarity criteria: AAA (or AA), SSS, SAS.

Part A: Multiple choice 16 questions

1. All circles are

  1. congruent
  2. similar
  3. neither similar nor congruent
  4. similar only if radii are equal

Every circle has the same shape.

2. Which pair of figures is always similar?

  1. Two rectangles
  2. Two rhombuses
  3. Two squares
  4. Two isosceles triangles

All angles are 90° and all sides are in the same ratio.

3. In ΔABC, DE ∥ BC with D on AB and E on AC. Then

  1. AD/DB = AE/EC
  2. AD/DB = EC/AE
  3. AD × DB = AE × EC
  4. AD + DB = AE + EC

This is the Basic Proportionality Theorem.

4. In ΔABC, DE ∥ BC, AD = 1.5 cm, DB = 3 cm and AE = 1 cm. EC is

  1. 1.5 cm
  2. 3 cm
  3. 2 cm
  4. 4.5 cm

1.5/3 = 1/EC, so EC = 2 cm.

5. DE ∥ BC, AD = 4 cm, DB = 6 cm, AE = 6 cm. EC is

  1. 4 cm
  2. 9 cm
  3. 8 cm
  4. 10 cm

4/6 = 6/EC, so EC = 9 cm.

6. By the AA criterion, two triangles are similar if

  1. two sides are equal
  2. one angle is equal
  3. all sides are equal
  4. two angles of one equal two angles of the other

The third angles are then equal automatically.

7. ΔABC ~ ΔPQR with AB/PQ = 1/2. If AB = 3 cm, PQ is

  1. 1.5 cm
  2. 6 cm
  3. 3 cm
  4. 9 cm

PQ = 2 × AB = 6 cm.

8. Which of these is NOT a similarity criterion for triangles?

  1. AA
  2. SSS
  3. SAS
  4. SSA

Two sides and a non-included angle do not fix the shape.

9. ΔABC ~ ΔDEF, ∠A = 47° and ∠E = 83°. ∠C is

  1. 50°
  2. 47°
  3. 83°
  4. 130°

∠B = ∠E = 83°, so ∠C = 180° − 47° − 83° = 50°.

10. A 6 m pole casts a 4 m shadow. At the same time a tower casts a 28 m shadow. The tower's height is

  1. 18 m
  2. 42 m
  3. 24 m
  4. 36 m

6/4 = h/28, so h = 42 m.

11. Two triangles are similar if their corresponding sides are

  1. equal
  2. in the same ratio
  3. parallel
  4. perpendicular

Proportional sides (SSS) give similarity.

12. Similar figures always have the same

  1. size
  2. area
  3. shape
  4. perimeter

Size may differ; shape stays the same.

13. D and E are midpoints of AB and AC of ΔABC. If DE = 5 cm, BC is

  1. 5 cm
  2. 2.5 cm
  3. 15 cm
  4. 10 cm

DE ∥ BC and DE = BC/2.

14. Triangles with sides 3, 4, 5 and 6, 8, 10 are

  1. similar by SSS
  2. not similar
  3. congruent
  4. similar by SAS only

All ratios equal 1/2.

15. If ΔABC ~ ΔQRP, then ∠B equals

  1. ∠Q
  2. ∠R
  3. ∠P
  4. ∠A

The order of letters gives the matching: A↔Q, B↔R, C↔P.

16. A line that divides two sides of a triangle in the same ratio is

  1. perpendicular to the third side
  2. equal to the third side
  3. parallel to the third side
  4. a median

This is the converse of the Basic Proportionality Theorem.

Part B: Fill in the blanks 8 blanks

17. Two triangles are similar if corresponding angles are [[equal]] and corresponding sides are [[proportional|in the same ratio|in proportion]].
18. The Basic Proportionality Theorem is also called [[Thales]] theorem.
19. All [[congruent]] figures are similar, but similar figures need not be congruent.
20. If two angles of one triangle equal two angles of another, the triangles are similar by the [[AA|AAA]] criterion.
21. In ΔABC, DE ∥ BC, AD = 2 cm, DB = 3 cm and AE = 4 cm. Then EC = [[6]] cm.
22. A 1.5 m student casts a 2 m shadow. At the same time a tree casts a 12 m shadow, so the tree is [[9]] m tall.
23. Two equilateral triangles of sides 3 cm and 5 cm are always [[similar]].

Part C: Questions and answers 9 questions

24. State the Basic Proportionality Theorem.

If a line is drawn parallel to one side of a triangle to meet the other two sides at distinct points, then the other two sides are divided in the same ratio.

25. In ΔABC, DE ∥ BC with AD/DB = 3/5 and AE = 4.5 cm. Find EC.

By BPT, AD/DB = AE/EC → 3/5 = 4.5/EC → EC = 4.5 × 5/3 = 7.5 cm.

26. E and F lie on sides PQ and PR of ΔPQR. PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm, FR = 2.4 cm. Is EF ∥ QR?

PE/EQ = 3.9/3 = 1.3 and PF/FR = 3.6/2.4 = 1.5.
The ratios are not equal, so EF is not parallel to QR.

27. In ΔPQR, PE = 4 cm, QE = 4.5 cm, PF = 8 cm, RF = 9 cm. Is EF ∥ QR?

PE/EQ = 4/4.5 = 8/9 and PF/FR = 8/9.
The ratios are equal, so by the converse of BPT, EF ∥ QR.

28. ABCD is a trapezium with AB ∥ DC, and its diagonals meet at O. Show that AO/BO = CO/DO.

In ΔAOB and ΔCOD: ∠OAB = ∠OCD and ∠OBA = ∠ODC (alternate angles, AB ∥ DC).
So ΔAOB ~ ΔCOD (AA). Hence AO/CO = BO/DO, which gives AO/BO = CO/DO.

29. A girl 90 cm tall walks away from a 3.6 m lamp post at 1.2 m/s. Find the length of her shadow after 4 seconds.

She is 4.8 m from the post. Let the shadow be s m.
Similar triangles: 0.9/s = 3.6/(4.8 + s) → 3.6s = 4.32 + 0.9s → 2.7s = 4.32 → s = 1.6 m.

30. A vertical pole 6 m long casts a 4 m shadow, while a tower casts a 28 m shadow at the same time. Find the height of the tower.

The sun's rays make the same angle, so the triangles are similar.
6/4 = h/28 → h = 42 m.

31. What is the difference between congruent and similar figures?

Congruent figures have the same shape and same size. Similar figures have the same shape but may have different sizes. Every congruent pair is similar, but not every similar pair is congruent.

32. In ΔABC, D and E are points on AB and AC with AD = 3 cm, AB = 9 cm, AE = 4 cm and AC = 12 cm. Is DE ∥ BC?

DB = 6 and EC = 8. AD/DB = 3/6 = 1/2 and AE/EC = 4/8 = 1/2.
The ratios are equal, so DE ∥ BC.

Part D: Word puzzles 4 puzzles

Newer lesson Older lesson