Similarity & Congruence - ACT Math

1. What is the hypotenuse-leg (HL) congruence theorem?

   Right triangles are congruent if hypotenuse and one leg are equal. Special case for right triangles only.

2. What is the Angle-Angle (AA) similarity criterion for triangles?

   Two triangles are similar if two angles are equal. Third angle is automatically equal by angle sum.

3. Identify the test for congruence in triangles.

   SSS, SAS, ASA, AAS, and HL for right triangles. These are the five triangle congruence postulates.

4. Find the new area: a figure has area $18$ and is dilated by $k=3$; what is the new area?

   $162$. New area is $18 \times 3^2 = 18 \times 9 = 162$.

5. Calculate the scale factor: $\triangle ABC \sim \triangle DEF$, $AB = 4$, $DE = 8$.

   Scale factor is $\frac{1}{2}$. Scale factor is $\frac{AB}{DE} = \frac{4}{8} = \frac{1}{2}$.

6. What must be true for two rectangles to be similar?

   Their corresponding sides must be proportional. Length-to-width ratios must be equal.

7. Which theorem states triangles are congruent if two sides and an included angle are equal?

   Side-Angle-Side (SAS) congruence theorem. Two sides and the angle between them.

8. What is the condition for two right triangles to be congruent?

   They are congruent if the hypotenuse and one leg are equal (HL theorem). HL is specific to right triangles.

9. Identify if triangles with angles 30°, 60°, 90° and 30°, 60°, 90° are similar.

   Yes, they are similar by AA similarity postulate. Identical angles guarantee similarity.

10. Identify the similarity criterion for polygons.

    Corresponding angles are equal and sides are proportional. General rule for all similar polygons.

11. What triangle congruence criterion is abbreviated $ASA$?

    Two angles and the included side are equal. Angle-Side-Angle: side must be between the two angles.

12. Identify the congruence criterion: $\triangle ABC \cong \triangle DEF$ if $\angle A = \angle D$, $\angle B = \angle E$, $AB = DE$.

    Angle-Side-Angle (ASA). Two angles and their included side match.

13. State the condition for two triangles to be similar.

    Corresponding angles are equal and sides are proportional. This ensures triangles have the same shape.

14. Find $x$: if $\triangle ABC \sim \triangle DEF$, $AB=9$, $DE=6$, and $BC=12$, what is $EF$?

    $8$. Scale factor is $\frac{6}{9} = \frac{2}{3}$, so $EF = 12 \times \frac{2}{3} = 8$.

15. What is the effect of a dilation with scale factor $k$ on a length $L$?

    New length is $kL$. All lengths multiply by the scale factor.

16. Find the length of a missing side: $\triangle XYZ \sim \triangle RST$, $XY = 5$, $RS = 10$, $XZ = 7$. Find $RT$.

    $RT = 14$. Scale factor is $\frac{10}{5} = 2$, so $RT = 7 \times 2 = 14$.

17. What does the congruence statement $\triangle ABC \cong \triangle DEF$ tell you about correspondence?

    $A\leftrightarrow D$, $B\leftrightarrow E$, $C\leftrightarrow F$. Order of vertices shows which parts correspond.

18. What triangle similarity criterion is abbreviated $SAS$ similarity?

    Included angle equal and adjacent sides in proportion. Side-Angle-Side for similarity uses proportional sides.

19. If $\triangle ABC \sim \triangle DEF$, and $\angle A = 30^\circ$, $\angle B = 60^\circ$, find $\angle E$.

    $\angle E = 60^\circ$. Corresponding angles are equal in similar triangles.

20. Identify the missing value: if similar triangles have $k=\frac{3}{2}$ and a side is $10$, what is the corresponding side?

    $15$. Multiply original side by scale factor: $10 \times \frac{3}{2} = 15$.

21. Which symbol denotes congruence between $\triangle ABC$ and $\triangle DEF$?

    $\triangle ABC \cong \triangle DEF$. The symbol $\cong$ denotes congruence.

22. What does it mean for two figures to be similar?

    Same shape; corresponding angles equal and sides proportional. Similar figures have the same shape but different sizes.

23. Identify the congruence criterion: $\triangle ABC \cong \triangle DEF$ if $AB = DE$, $BC = EF$, $CA = FD$.

    Side-Side-Side (SSS). All three pairs of sides are equal.

24. What is the effect of similarity on the areas of two triangles?

    The ratio of the areas is the square of the ratio of corresponding sides. Area scales with the square of linear scale factor.

25. Identify the Side-Angle-Side (SAS) similarity criterion for triangles.

    Two sides in proportion and the included angle equal. Ensures proportional scaling with fixed angle.

26. What is the key difference between similar and congruent figures?

    Similar figures have proportional sides, congruent figures have equal sides and angles. Similarity allows proportional scaling; congruence requires exactness.

27. What is the formula for finding a missing side in similar triangles?

    Use the proportion: $\frac{a}{b} = \frac{c}{d}$. Cross-multiply to solve for unknown side.

28. What is the Side-Angle-Side (SAS) similarity criterion in triangle similarity?

    Two sides are proportional and the included angle is equal. Proportional sides with equal included angle.

29. Does the AAA criterion prove triangle congruence?

    No, AAA only proves similarity, not congruence. AAA doesn't guarantee equal side lengths.

30. What is the scale factor from figure $A$ to figure $B$ in a similarity?

    $k=\frac{\text{side in }B}{\text{corresponding side in }A}$. Scale factor is the ratio of corresponding lengths.

31. What is the Angle-Side-Angle (ASA) congruence criterion?

    Two angles and the included side are equal. Two angles and their included side determine congruence.

32. Identify whether $SSS$ similarity applies: sides are $3,4,5$ and $6,8,10$; are the triangles similar?

    Yes, by $SSS$ similarity with $k=2$. All sides proportional with ratio $\frac{6}{3} = 2$.

33. Which triangles are congruent by the ASA postulate?

    Triangles with two equal angles and an included equal side. ASA requires the side between the two angles.

34. Find the new perimeter: a figure has perimeter $28$ and is dilated by $k=\frac{5}{2}$; what is the new perimeter?

    $70$. New perimeter is $28 \times \frac{5}{2} = 70$.

35. What right-triangle congruence criterion is abbreviated $HL$?

    Hypotenuse and one leg are equal in two right triangles. Hypotenuse-Leg: specific to right triangles only.

36. Identify whether $AA$ similarity applies: one triangle has angles $40^\circ,60^\circ,80^\circ$ and another has $40^\circ,60^\circ,80^\circ$.

    Yes, by $AA$ similarity. Two pairs of angles are equal.

37. What does it mean if two polygons are similar?

    Corresponding angles are equal and sides are proportional. These are the defining properties of similarity.

38. What is the value of the scale factor $k$ for congruent figures?

    $k=1$. Congruent means scale factor equals one.

39. What is the criterion for Side-Angle-Side (SAS) congruence?

    Two sides and the included angle are equal. Two sides and their included angle determine congruence.

40. What is the volume relationship for similar solids with scale factor $k$?

    Volumes scale by $k^3$. Volume scales by the cube of the linear factor.

41. If two triangles are congruent, are they also similar?

    Yes, congruent triangles are always similar. Congruence is a special case of similarity.

42. What is the definition of similar triangles?

    Triangles with corresponding angles equal and sides proportional. This defines similarity: shape preserved, size scaled.

43. Calculate the side ratio: $\triangle ABC \sim \triangle DEF$, $AB = 10$, $DE = 20$.

    Ratio is $1:2$. Side ratio is $\frac{AB}{DE} = \frac{10}{20} = \frac{1}{2}$.

44. Find the linear scale factor: if $\frac{A_2}{A_1}=49$ for similar figures, what is $k$?

    $7$. Take square root of area ratio: $\sqrt{49} = 7$.

45. Find the missing side: In $\triangle ABC \sim \triangle DEF$, $AB = 6$, $BC = 8$, $EF = 12$. Find $DE$.

    $DE = 9$. Scale factor is $\frac{12}{8} = \frac{3}{2}$, so $DE = 6 \times \frac{3}{2} = 9$.

46. Which transformation(s) always preserve distance and angle measure (rigid motions)?

    Translations, rotations, and reflections. Rigid motions preserve both size and shape.

47. What triangle congruence criterion is abbreviated $SSS$?

    All $3$ pairs of corresponding sides are equal. Side-Side-Side: all three sides match exactly.

48. Determine if triangles with sides 3, 4, 5 and 6, 8, 10 are similar.

    Yes, they are similar by SSS similarity theorem. Sides are in ratio $2:1$, proving similarity.

49. What is the ratio of areas of two similar triangles if the ratio of their sides is $3:5$?

    $\frac{9}{25}$. Area ratio equals $(\frac{3}{5})^2 = \frac{9}{25}$

50. Find the linear scale factor: if $\frac{V_2}{V_1}=\frac{27}{8}$ for similar solids, what is $k$?

    $\frac{3}{2}$. Take cube root of volume ratio: $\sqrt[3]{\frac{27}{8}} = \frac{3}{2}$.

51. Which property is true for corresponding angles of similar figures?

    They are equal. Angle measures remain unchanged in similar figures.

52. What is the perimeter relationship for similar figures with scale factor $k$?

    Perimeters scale by $k$. Linear dimensions scale by the scale factor.

53. Identify the criterion: $\triangle ABC \cong \triangle DEF$ if $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$.

    Angle-Angle-Angle (AAA) is not a congruence criterion. Equal angles alone don't guarantee equal size.

54. What is the condition for Side-Side-Side (SSS) congruence?

    All three sides are equal in length. All three pairs of sides must match exactly.

55. What is the definition of similar figures?

    Figures with the same shape but not necessarily the same size. Angles match but sides can be scaled proportionally.

56. How does congruence differ from similarity?

    Congruence requires equal sides and angles; similarity requires proportional sides and equal angles. Similarity allows scaling; congruence requires exact match.

57. Identify the scale factor: a side changes from $12$ to $18$ under similarity; what is $k$?

    $\frac{3}{2}$. Divide new length by original: $\frac{18}{12} = \frac{3}{2}$.

58. Calculate the area ratio: $\triangle ABC \sim \triangle DEF$, side ratio $2:5$.

    Area ratio is $\frac{4}{25}$. Area ratio is $(\frac{2}{5})^2 = \frac{4}{25}$.

59. Find the area scale factor: if the linear scale factor is $k=4$, what is $\frac{A_2}{A_1}$?

    $16$. Area scale factor is linear factor squared: $4^2 = 16$.

60. Find the perimeter of $\triangle DEF$: $\triangle ABC \sim \triangle DEF$, perimeter of $\triangle ABC = 24$, scale factor $\frac{1}{3}$.

    Perimeter of $\triangle DEF = 8$. Perimeter scales by the same factor as sides.

61. In similar polygons, the ratio of perimeters is equal to what?

    The ratio of their corresponding side lengths. Perimeters scale linearly with side lengths.

62. Which side is the hypotenuse in a right triangle used in $HL$ congruence?

    The side opposite the right angle. Longest side in a right triangle.

63. Given similar triangles, what is the ratio of their circumferences?

    The ratio of circumferences equals the ratio of corresponding sides. Linear measurements scale proportionally.

64. What does the similarity statement $\triangle ABC \sim \triangle DEF$ tell you about correspondence?

    $A\leftrightarrow D$, $B\leftrightarrow E$, $C\leftrightarrow F$. Order of vertices shows which parts correspond.

65. Identify the correct conclusion: if $\triangle ABC \cong \triangle DEF$, what is $\frac{AB}{DE}$?

    $1$. Congruent figures have scale factor $k = 1$.

66. What is the relationship between corresponding angles in similar triangles?

    Corresponding angles are equal in measure. Similar triangles preserve all angle measures.

67. What is the formula for the ratio of areas of two similar triangles?

    The square of the ratio of their corresponding sides. Area scales as the square of linear scale factor.

68. Determine the missing side: $\triangle ABC \sim \triangle DEF$, $AB = 3$, $BC = 4$, $AC = 5$, $DE = 6$. Find $EF$.

    $EF = 8$. Scale factor is $\frac{6}{3} = 2$, so $EF = 4 \times 2 = 8$.

69. What is the Angle-Side-Angle (ASA) congruence postulate?

    Triangles are congruent if two angles and the included side are equal. Two angles and the side between them.

70. Are equilateral triangles always similar?

    Yes, all equilateral triangles are similar. All angles are $60°$, so AA applies.

71. What triangle similarity criterion is abbreviated $SSS$ similarity?

    All $3$ pairs of corresponding sides are proportional. Side-Side-Side for similarity uses proportional ratios.

72. Find $x$: if similar triangles satisfy $\frac{AB}{DE}=\frac{4}{7}$ and $DE=21$, what is $AB$?

    $12$. Cross multiply: $AB = DE \times \frac{4}{7} = 21 \times \frac{4}{7} = 12$.

73. What is the ratio of corresponding sides in similar triangles?

    The ratio is constant for all corresponding sides. Scale factor remains consistent throughout.

74. Find $x$: if $\triangle ABC \sim \triangle DEF$ and $\frac{AB}{DE}=\frac{2}{5}$ with $AB=8$, what is $DE$?

    $20$. Cross multiply: $\frac{8}{DE} = \frac{2}{5}$, so $DE = 20$.

75. What does it mean for two figures to be congruent?

    Same shape and size; all corresponding sides and angles are equal. Congruent figures are identical copies.

76. What triangle congruence criterion is abbreviated $SAS$?

    Two sides and the included angle are equal. Side-Angle-Side: angle must be between the two sides.

77. Find the length of the missing side: $\triangle ABC \sim \triangle DEF$, $AB = 9$, $BC = 12$, $EF = 16$. Find $DE$.

    $DE = 12$. Scale factor is $\frac{16}{12} = \frac{4}{3}$, so $DE = 9 \times \frac{4}{3} = 12$.

78. Find $x$: if similar figures have perimeters $P_1=30$ and $P_2=45$, what is the scale factor $k=\frac{P_2}{P_1}$?

    $\frac{3}{2}$. Perimeter ratio equals linear scale factor: $\frac{45}{30} = \frac{3}{2}$.

79. State the definition of the Hypotenuse-Leg (HL) congruence criterion.

    In right triangles, hypotenuse and one leg are equal. Special congruence criterion for right triangles only.

80. What is the area relationship for similar figures with scale factor $k$?

    Areas scale by $k^2$. Area scales by the square of the linear factor.

81. Which transformation changes size but preserves shape and angles?

    A dilation. Dilations change size while preserving shape.

82. What is the Side-Side-Side (SSS) similarity criterion for triangles?

    Three sides in one triangle are proportional to three sides in another. All sides scaled by the same factor.

83. Are two isosceles triangles always similar?

    No, isosceles triangles are not always similar. Different base angles prevent guaranteed similarity.

84. Find the volume scale factor: if the linear scale factor is $k=\frac{1}{2}$, what is $\frac{V_2}{V_1}$?

    $\frac{1}{8}$. Volume scale factor is linear factor cubed: $(\frac{1}{2})^3 = \frac{1}{8}$.

85. How do you denote that two triangles $\triangle ABC$ and $\triangle DEF$ are similar?

    $\triangle ABC \sim \triangle DEF$. The symbol $\sim$ denotes similarity.

86. State the similarity condition based on proportional sides.

    Corresponding sides are proportional. Equal ratios between corresponding sides.

87. What triangle similarity criterion is abbreviated $AA$?

    Two pairs of corresponding angles are equal. Angle-Angle: third angle is automatically equal.

88. Identify whether $SSS$ congruence applies: sides are $5,7,9$ and $5,7,9$; are the triangles congruent?

    Yes, by $SSS$. All three pairs of sides are equal.

89. Can right triangles be similar to non-right triangles?

    No, a right triangle cannot be similar to a non-right triangle. Different angle sets prevent similarity.

90. State the Angle-Angle (AA) similarity postulate.

    Two triangles are similar if two angles are equal. Two equal angles ensure the third is also equal.

91. Which theorem states that triangles are congruent if all sides are equal?

    Side-Side-Side (SSS) congruence theorem. All three sides equal proves congruence.

92. If two triangles are similar, what can be said about their perimeters?

    Perimeters are in the same ratio as their corresponding sides. Perimeter scales with linear dimensions.

93. Are triangles with sides 5, 12, 13 and 10, 24, 26 congruent?

    No, they are only similar but not congruent. Different sizes prevent congruence.

94. What is the Hypotenuse-Leg (HL) congruence criterion specific to?

    Right triangles. Only applies to triangles with right angles.

95. Which is true for the corresponding sides of congruent triangles?

    They are equal. Corresponding sides have identical lengths.

96. What is the definition of congruent figures?

    Figures with equal size and shape. All corresponding angles and sides match exactly.

97. What is the relationship between corresponding sides in similar triangles?

    Corresponding side lengths are proportional. All ratios of corresponding sides are equal.

98. State the Side-Angle-Side (SAS) similarity theorem.

    Two triangles are similar if two sides are proportional and the included angle is equal. Two proportional sides with equal included angle.

99. What is the Angle-Angle-Side (AAS) congruence theorem?

    Triangles are congruent if two angles and a non-included side are equal. Two angles and any non-included side.

100. Find the missing angle: if $\triangle ABC \sim \triangle DEF$ and $\angle A=35^\circ$, what is $\angle D$?

     $35^\circ$. Corresponding angles in similar triangles are equal.

101. In $\triangle ABC \cong \triangle DEF$, if $AB = 5$, $BC = 7$, find $DE$ and $EF$.

     $DE = 5$, $EF = 7$. Congruent triangles have equal corresponding sides.

102. What is the Angle-Angle-Side (AAS) congruence criterion?

     Two angles and a non-included side are equal. Two angles and a non-included side determine congruence.

103. What is the definition of congruent triangles?

     Triangles with exactly equal corresponding sides and angles. Congruence means identical in size and shape.

104. What is the condition for Angle-Angle (AA) similarity?

     Two angles of one triangle are equal to two angles of another. Two equal angles ensure the third is also equal.

105. What is the Side-Side-Side (SSS) similarity theorem?

     Triangles are similar if corresponding sides are proportional. All sides in same ratio proves similarity.

106. Find the missing side of a similar triangle with sides 3, 4, and ? given similar triangle sides 6, 8, 10.

     The missing side is 5. Use proportional sides: $\frac{3}{6} = \frac{?}{10}$.

107. What triangle congruence criterion is abbreviated $AAS$?

     Two angles and a non-included side are equal. Angle-Angle-Side: side is not between the angles.

108. What is the basic property of corresponding angles in similar triangles?

     They are equal. Angle measures are preserved in similar figures.

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