ACT Math Flashcards: Similarity And Congruence

What this deck covers

This deck focuses on Similarity And Congruence, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

Flashcard 2: Are equilateral triangles always similar?

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

Flashcard 3: Identify the criterion: $\triangle ABC \cong \triangle DEF$ if $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$.

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

Flashcard 4: State the condition for two triangles to be similar.

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

Flashcard 5: What is the Angle-Angle (AA) similarity criterion for triangles?

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

Flashcard 6: What is the basic property of corresponding angles in similar triangles?

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

Flashcard 7: Which side is the hypotenuse in a right triangle used in $HL$ congruence?

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

Flashcard 8: Which triangles are congruent by the ASA postulate?

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

Flashcard 9: Which transformation changes size but preserves shape and angles?

Answer: A dilation. Dilations change size while preserving shape.

Flashcard 10: Find $x$: if $\triangle ABC \sim \triangle DEF$ and $\frac{AB}{DE}=\frac{2}{5}$ with $AB=8$, what is $DE$?

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

Flashcard 11: What triangle congruence criterion is abbreviated $AAS$?

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

Flashcard 12: What does it mean for two figures to be congruent?

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

Flashcard 13: What must be true for two rectangles to be similar?

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

Flashcard 14: Identify the test for congruence in triangles.

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

Flashcard 15: Identify if triangles with angles 30°, 60°, 90° and 30°, 60°, 90° are similar.

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

Flashcard 16: What right-triangle congruence criterion is abbreviated $HL$?

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

Flashcard 17: What is the criterion for Side-Angle-Side (SAS) congruence?

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

Flashcard 18: Identify whether $SSS$ congruence applies: sides are $5,7,9$ and $5,7,9$; are the triangles congruent?

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

Flashcard 19: Find the new area: a figure has area $18$ and is dilated by $k=3$; what is the new area?

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

Flashcard 20: Calculate the scale factor: $\triangle ABC \sim \triangle DEF$, $AB = 4$, $DE = 8$.

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

Flashcard 21: What must be true for two rectangles to be similar?

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

Flashcard 22: Which theorem states triangles are congruent if two sides and an included angle are equal?

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

Flashcard 23: State the Side-Angle-Side (SAS) similarity theorem.

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

Flashcard 24: What is the condition for two right triangles to be congruent?

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

Flashcard 25: Identify if triangles with angles 30°, 60°, 90° and 30°, 60°, 90° are similar.

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

Flashcard 26: Identify the similarity criterion for polygons.

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

Flashcard 27: What is the relationship between corresponding sides in similar triangles?

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

Flashcard 28: What triangle congruence criterion is abbreviated $ASA$?

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

Flashcard 29: Identify the congruence criterion: $\triangle ABC \cong \triangle DEF$ if $\angle A = \angle D$, $\angle B = \angle E$, $AB = DE$.

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

Flashcard 30: What is the definition of congruent triangles?

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

Flashcard 31: State the condition for two triangles to be similar.

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

Flashcard 32: Calculate the side ratio: $\triangle ABC \sim \triangle DEF$, $AB = 10$, $DE = 20$.

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

Flashcard 33: Find $x$: if $\triangle ABC \sim \triangle DEF$, $AB=9$, $DE=6$, and $BC=12$, what is $EF$?

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

Flashcard 34: What is the effect of a dilation with scale factor $k$ on a length $L$?

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

Flashcard 35: Find the length of a missing side: $\triangle XYZ \sim \triangle RST$, $XY = 5$, $RS = 10$, $XZ = 7$. Find $RT$.

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

Flashcard 36: If $\triangle ABC \sim \triangle DEF$, and $\angle A = 30^\circ$, $\angle B = 60^\circ$, find $\angle E$.

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

Flashcard 37: Find the perimeter of $\triangle DEF$: $\triangle ABC \sim \triangle DEF$, perimeter of $\triangle ABC = 24$, scale factor $\frac{1}{3}$.

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

Flashcard 38: What does the congruence statement $\triangle ABC \cong \triangle DEF$ tell you about correspondence?

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

Flashcard 39: What triangle similarity criterion is abbreviated $SAS$ similarity?

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

Flashcard 40: If $\triangle ABC \sim \triangle DEF$, and $\angle A = 30^\circ$, $\angle B = 60^\circ$, find $\angle E$.

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

Flashcard 41: In $\triangle ABC \cong \triangle DEF$, if $AB = 5$, $BC = 7$, find $DE$ and $EF$.

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

Flashcard 42: Identify the missing value: if similar triangles have $k=\frac{3}{2}$ and a side is $10$, what is the corresponding side?

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

Flashcard 43: Which symbol denotes congruence between $\triangle ABC$ and $\triangle DEF$?

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

Flashcard 44: What does it mean for two figures to be similar?

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

Flashcard 45: Identify the congruence criterion: $\triangle ABC \cong \triangle DEF$ if $AB = DE$, $BC = EF$, $CA = FD$.

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

Flashcard 46: What is the effect of similarity on the areas of two triangles?

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

Flashcard 47: Identify the Side-Angle-Side (SAS) similarity criterion for triangles.

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

Flashcard 48: What is the Hypotenuse-Leg (HL) congruence criterion specific to?

Answer: Right triangles. Only applies to triangles with right angles.

Flashcard 49: What is the key difference between similar and congruent figures?

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

Flashcard 50: What is the definition of similar figures?

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

Flashcard 51: What is the formula for finding a missing side in similar triangles?

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

Flashcard 52: What is the Angle-Angle-Side (AAS) congruence theorem?

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

Flashcard 53: What is the Side-Angle-Side (SAS) similarity criterion in triangle similarity?

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

Flashcard 54: Find the linear scale factor: if $\frac{V_2}{V_1}=\frac{27}{8}$ for similar solids, what is $k$?

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

Flashcard 55: Does the AAA criterion prove triangle congruence?

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

Flashcard 56: Given similar triangles, what is the ratio of their circumferences?

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

Flashcard 57: What is the scale factor from figure $A$ to figure $B$ in a similarity?

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

Flashcard 58: Which transformation(s) always preserve distance and angle measure (rigid motions)?

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

Flashcard 59: What is the Angle-Side-Angle (ASA) congruence criterion?

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

Flashcard 60: What does it mean if two polygons are similar?

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

Flashcard 61: Identify whether $SSS$ similarity applies: sides are $3,4,5$ and $6,8,10$; are the triangles similar?

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

Flashcard 62: Which triangles are congruent by the ASA postulate?

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

Flashcard 63: Find the new perimeter: a figure has perimeter $28$ and is dilated by $k=\frac{5}{2}$; what is the new perimeter?

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

Flashcard 64: Find $x$: if similar figures have perimeters $P_1=30$ and $P_2=45$, what is the scale factor $k=\frac{P_2}{P_1}$?

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

Flashcard 65: What right-triangle congruence criterion is abbreviated $HL$?

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

Flashcard 66: How do you denote that two triangles $\triangle ABC$ and $\triangle DEF$ are similar?

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

Flashcard 67: Identify whether $AA$ similarity applies: one triangle has angles $40^\circ,60^\circ,80^\circ$ and another has $40^\circ,60^\circ,80^\circ$.

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

Flashcard 68: What does it mean if two polygons are similar?

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

Flashcard 69: Identify the congruence criterion: $\triangle ABC \cong \triangle DEF$ if $AB = DE$, $BC = EF$, $CA = FD$.

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

Flashcard 70: Find the length of the missing side: $\triangle ABC \sim \triangle DEF$, $AB = 9$, $BC = 12$, $EF = 16$. Find $DE$.

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

Flashcard 71: What is the value of the scale factor $k$ for congruent figures?

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

Flashcard 72: What is the criterion for Side-Angle-Side (SAS) congruence?

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

Flashcard 73: What is the volume relationship for similar solids with scale factor $k$?

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

Flashcard 74: If two triangles are congruent, are they also similar?

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

Flashcard 75: Find the volume scale factor: if the linear scale factor is $k=\frac{1}{2}$, what is $\frac{V_2}{V_1}$?

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

Flashcard 76: What triangle congruence criterion is abbreviated $SAS$?

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

Flashcard 77: What is the definition of similar triangles?

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

Flashcard 78: State the Angle-Angle (AA) similarity postulate.

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

Flashcard 79: Calculate the side ratio: $\triangle ABC \sim \triangle DEF$, $AB = 10$, $DE = 20$.

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

Flashcard 80: What is the scale factor from figure $A$ to figure $B$ in a similarity?

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

Flashcard 81: Find the linear scale factor: if $\frac{A_2}{A_1}=49$ for similar figures, what is $k$?

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

Flashcard 82: What is the condition for Side-Side-Side (SSS) congruence?

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

Flashcard 83: Find the missing side: In $\triangle ABC \sim \triangle DEF$, $AB = 6$, $BC = 8$, $EF = 12$. Find $DE$.

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

Flashcard 84: What is the volume relationship for similar solids with scale factor $k$?

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

Flashcard 85: Find the new perimeter: a figure has perimeter $28$ and is dilated by $k=\frac{5}{2}$; what is the new perimeter?

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

Flashcard 86: Which transformation(s) always preserve distance and angle measure (rigid motions)?

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

Flashcard 87: What triangle congruence criterion is abbreviated $SSS$?

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

Flashcard 88: Determine if triangles with sides 3, 4, 5 and 6, 8, 10 are similar.

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

Flashcard 89: What is the ratio of areas of two similar triangles if the ratio of their sides is $3:5$?

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

Flashcard 90: Find the linear scale factor: if $\frac{V_2}{V_1}=\frac{27}{8}$ for similar solids, what is $k$?

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

Flashcard 91: Which property is true for corresponding angles of similar figures?

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

Flashcard 92: What is the effect of similarity on the areas of two triangles?

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

Flashcard 93: What is the perimeter relationship for similar figures with scale factor $k$?

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

Flashcard 94: Identify the criterion: $\triangle ABC \cong \triangle DEF$ if $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$.

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

Flashcard 95: What is the Angle-Angle-Side (AAS) congruence criterion?

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

Flashcard 96: Identify whether $AA$ similarity applies: one triangle has angles $40^\circ,60^\circ,80^\circ$ and another has $40^\circ,60^\circ,80^\circ$.

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

Flashcard 97: Are two isosceles triangles always similar?

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

Flashcard 98: What is the condition for Side-Side-Side (SSS) congruence?

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

Flashcard 99: What is the definition of similar figures?

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

Flashcard 100: What is the Angle-Side-Angle (ASA) congruence postulate?

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