SIMILARITIES REVISION QUESTIONS WITH ANSWERS

Triangle Similarity Questions

1. Define what it means for two triangles to be similar.

2. Triangle \(ABC\) has angles \(60^\circ\), \(60^\circ\), and \(60^\circ\). Triangle \(DEF\) also has the same angles. Are they similar?

3. What are the three criteria for triangle similarity?

4. Two triangles have sides in the ratio 3:4:5 and 6:8:10. Are they similar?

5. How does the AA (Angle-Angle) criterion prove triangle similarity?

6. Are all equilateral triangles similar?

7. Triangle \(XYZ\) and triangle \(PQR\) have all corresponding angles equal. Are they similar?

8. Two right triangles have one acute angle in common. Are they similar?

9. Prove that two triangles with sides in proportion and included angle equal are not necessarily similar.

10. Explain how triangle similarity differs from triangle congruence.

11. What is the effect of enlargement (dilation) on triangle similarity?

12. Triangle \(ABC\) has sides 5, 6, and 7. Triangle \(DEF\) has sides 10, 12, and 14. Are they similar?

13. Can AAA be used for proving triangle similarity?

14. Can two triangles be similar if they share only one equal angle?

15. Two triangles have sides in the ratio 2:3. What does this imply about their perimeters and areas?

16. In the figure below, identify whether the two triangles are similar.

17. Triangle \(ABC\) is similar to triangle \(DEF\). If \(AB = 3\) cm and \(DE = 6\) cm, what is the scale factor?

18. List three properties that remain unchanged in similar triangles.

19. Prove that a triangle and its enlargement are always similar.

20. Explain how to use corresponding angles to prove similarity.

21. In triangle \(PQR\) and triangle \(XYZ\), \(PQ = 5\), \(QR = 8\), \(PR = 7\); \(XY = 10\), \(YZ = 16\), \(XZ = 14\). Are the triangles similar?

22. If two triangles are similar, are their altitudes in the same ratio as the sides?

23. A triangle has sides 4 cm, 6 cm, and 8 cm. Another triangle has sides 8 cm, 12 cm, and 16 cm. Are they similar?

24. Is it possible for two triangles to be similar but not congruent?

25. Two isosceles triangles have equal vertex angles. Are they similar?

26. Triangle \(ABC\) has angles \(A = 45^\circ\), \(B = 45^\circ\), \(C = 90^\circ\). Triangle \(DEF\) has the same angles. Are they similar?

27. Explain the SAS (Side-Angle-Side) criterion for similarity.

28. Are triangles with sides in ratio 2:2:3 and 4:4:6 similar?

29. Two right triangles have equal hypotenuse and one leg in same ratio. Are they similar?

30. Can triangle similarity be used in indirect measurement? Give an example.

Answers

1. Same shape, proportional sides, equal angles.
2. Yes, same angles = similar.
3. AA, SSS (in ratio), SAS (in ratio and angle).
4. Yes, sides are proportional.
5. Two equal angles guarantee similarity.
6. Yes.
7. Yes, AA rule.
8. Yes, both right and have equal angle.
9. No, angle must be between the proportional sides.
10. Similarity = shape; Congruence = shape + size.
11. Enlargement keeps shape, changes size.
12. Yes, sides in 1:2 ratio.
13. Yes, AAA proves similarity.
14. No, need at least 2 equal angles.
15. Perimeter scales with ratio, area with square of ratio.
16. Yes, angles equal = similar.
17. Scale factor = \(6 \div 3 = 2\).
18. Angles, shape, and internal ratios.
19. Yes, by enlargement definition.
20. Compare two equal angles.
21. Yes, sides in 1:2 ratio.
22. Yes.
23. Yes, 1:2 ratio.
24. Yes, similarity doesn't require same size.
25. Yes, base angles equal.
26. Yes.
27. SAS = 2 sides in same ratio and included angle equal.
28. Yes, 2:3 same ratio as 4:6.
29. Yes, use RHS with ratios.
30. Yes. Example: shadow problems or height of tall objects.