Triangles · Class 10 Mathematics: 60 flashcards
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1. When are two figures similar?
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When they have the same shape, whether or not they have the same size.
2. What is the difference between congruent and similar?
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Congruent means same shape AND same size; similar allows the size to differ.
3. Are all congruent figures similar?
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Yes — congruence is similarity with scale factor 1.
4. Are all similar figures congruent?
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No. The converse is false.
5. State the two conditions for two polygons to be similar.
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Corresponding angles equal, AND corresponding sides in the same ratio.
6. Why is a square not similar to a rectangle 4 by 7?
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The angles match but the side ratios $\frac44$ and $\frac47$ differ.
7. Why is a square not similar to a rhombus of the same side?
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The side ratios are all 1 but the angles differ.
8. Are all circles similar?
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Yes.
9. Are all squares similar?
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Yes.
10. Are all rectangles similar?
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No — a 2 × 3 and a 2 × 9 rectangle are not the same shape.
11. What does $ABCD \sim PQRS$ tell you about the pairing?
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A matches P, B matches Q, C matches R, D matches S.
12. Write the side ratios for $ABCD \sim PQRS$.
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$\frac{AB}{PQ} = \frac{BC}{QR} = \frac{CD}{RS} = \frac{DA}{SP}$.
13. What is the scale factor?
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The common value of the ratios of corresponding sides.
14. What does a scale factor of 1 mean?
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The figures are congruent.
15. $\triangle ABC \sim \triangle PQR$ with $AB = 6$, $PQ = 9$, $BC = 8$. Find $QR$.
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$\frac69 = \frac8{QR}$, so $QR = 12$.
16. Two similar quadrilaterals have sides 4, 6, 8, 10 and 6, 9, 12, x. Find x.
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The scale factor is $\frac23$, so $x = 15$.
17. What is the symbol for 'is similar to'?
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$\sim$
18. What is the commonest error when setting up ratios?
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Flipping one fraction so the figures are not consistently on the same side.
19. What must a full 'are these similar?' answer contain?
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Both conditions named, each tested with numbers, and the verdict — naming the condition that failed.
20. Why is this chapter about triangles rather than polygons?
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For triangles one condition forces the other, so much less checking is needed.
21. State the Basic Proportionality Theorem.
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If a line runs parallel to one side of a triangle and meets the other two sides at two separate points, then those two sides are divided in the same ratio.
22. If $DE \parallel BC$ in triangle $ABC$, write the BPT in symbols.
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$\frac{AD}{DB} = \frac{AE}{EC}$
23. Write the part-over-whole form of the BPT.
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$\frac{AD}{AB} = \frac{AE}{AC}$
24. Why must you not write $\frac{AD}{DB} = \frac{AE}{AC}$?
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It mixes part-over-part with part-over-whole; the two sides must use the same form.
25. What is the construction in the BPT proof?
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Join BE and CD, then draw $DM \perp AC$ and $EN \perp AB$.
26. In the proof, what is $\frac{ar(ADE)}{ar(BDE)}$?
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$\frac{AD}{DB}$ — the common height EN cancels.
27. In the proof, what is $\frac{ar(ADE)}{ar(DEC)}$?
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$\frac{AE}{EC}$ — the common height DM cancels.
28. Why does $ar(BDE) = ar(DEC)$?
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They stand on the same base DE and lie between the same parallels DE and BC.
29. What must you write alongside $ar(BDE) = ar(DEC)$?
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The reason: same base, between the same parallels.
30. Does the BPT proof use similar triangles?
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No — only areas. Using similarity would be circular.
31. State the converse of the BPT.
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If a line cuts two sides of a triangle in equal proportion, that line runs parallel to the remaining side.
32. When do you use the BPT rather than its converse?
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When you are TOLD the line is parallel and need a ratio.
33. When do you use the converse?
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When you must SHOW that a line is parallel, by checking ratios.
34. $DE \parallel BC$ with $AD = 4$, $DB = 6$, $AE = 6$. Find EC.
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$\frac46 = \frac6{EC}$, so $EC = 9$.
35. $AD = 3$, $DB = 5$, $AE = 6$, $EC = 10$. Is $DE \parallel BC$?
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Yes — $\frac35 = \frac6{10}$, so by the converse it is parallel.
36. $AD = 2$, $DB = 3$, $AE = 4$, $EC = 5$. Is $DE \parallel BC$?
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No — $\frac23 \ne \frac45$.
37. What condition on the points appears in the BPT statement?
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The line must cut the other two sides in DISTINCT points.
38. What four headings should a proof answer use?
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Given, To prove, Construction, Proof.
39. Who is the theorem named after?
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Thales of Miletus.
40. How do you get from $\frac{AD}{DB} = \frac{AE}{EC}$ to the other form?
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Add 1 to both sides, giving $\frac{AB}{DB} = \frac{AC}{EC}$, then invert.
41. State the AAA similarity criterion.
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If corresponding angles of two triangles are equal, the triangles are similar.
42. State the AA similarity criterion.
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If two angles of one triangle equal two angles of another, the triangles are similar.
43. Why is AA enough?
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The angles add to 180°, so once two pairs match the third pair must match too.
44. State the SSS similarity criterion.
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If corresponding sides of two triangles are in the same ratio, the triangles are similar.
45. Are 3, 4, 5 and 6, 8, 10 similar?
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Yes — all three ratios equal $\frac12$.
46. Are 3, 4, 5 and 6, 8, 11 similar?
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No — the third ratio $\frac5{11}$ differs from $\frac12$.
47. State the SAS similarity criterion.
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If one angle of a triangle equals one angle of another and the sides INCLUDING those angles are proportional, the triangles are similar.
48. What does 'including' mean in SAS?
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The equal angle must lie between the two sides whose ratio is checked.
49. Which criterion fits two pairs of equal angles?
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AA.
50. Which criterion fits three pairs of proportional sides?
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SSS.
51. Which criterion fits one equal angle with the two sides around it proportional?
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SAS.
52. One angle equal and two sides proportional, but the angle is not between them. Which criterion applies?
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None of them — SAS requires the included angle.
53. What does $\triangle ABC \sim \triangle PQR$ assert about vertices?
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A matches P, B matches Q and C matches R.
54. If $\angle A = \angle Q$, $\angle B = \angle R$, $\angle C = \angle P$, how do you write the similarity?
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$\triangle ABC \sim \triangle QRP$.
55. $\triangle ABC \sim \triangle PQR$ with $AB = 6$, $BC = 9$, $PQ = 4$. Find QR.
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$\frac64 = \frac9{QR}$, so $QR = 6$.
56. Where do the three similarity criteria come from?
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They are proved from the Basic Proportionality Theorem.
57. What is the RHS similarity criterion?
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If the hypotenuse and one side of one right triangle are proportional to those of another, the triangles are similar.
58. How do the similarity criteria differ from the congruence criteria?
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Congruence needs EQUAL sides; similarity needs sides IN THE SAME RATIO.
59. Do similar triangles have equal sides?
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No — only their ratios are equal, unless the scale factor is 1.
60. What three things should a similarity answer contain?
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The equal angles or ratios with reasons, the criterion named, and the similarity statement in the right order.
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