Similarity & Congruence - ACT Math
Card 1 of 30
Are equilateral triangles always similar?
Are equilateral triangles always similar?
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Yes, all equilateral triangles are similar. All angles are $60°$, so AA applies.
Yes, all equilateral triangles are similar. All angles are $60°$, so AA applies.
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Which triangles are congruent by the ASA postulate?
Which triangles are congruent by the ASA postulate?
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Triangles with two equal angles and an included equal side. ASA requires the side between the two angles.
Triangles with two equal angles and an included equal side. ASA requires the side between the two angles.
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If two triangles are congruent, are they also similar?
If two triangles are congruent, are they also similar?
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Yes, congruent triangles are always similar. Congruence is a special case of similarity.
Yes, congruent triangles are always similar. Congruence is a special case of similarity.
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What is the Angle-Angle-Side (AAS) congruence theorem?
What is the Angle-Angle-Side (AAS) congruence theorem?
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Triangles are congruent if two angles and a non-included side are equal. Two angles and any non-included side.
Triangles are congruent if two angles and a non-included side are equal. Two angles and any non-included side.
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Which theorem states that triangles are congruent if all sides are equal?
Which theorem states that triangles are congruent if all sides are equal?
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Side-Side-Side (SSS) congruence theorem. All three sides equal proves congruence.
Side-Side-Side (SSS) congruence theorem. All three sides equal proves congruence.
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What triangle congruence criterion is abbreviated $ASA$?
What triangle congruence criterion is abbreviated $ASA$?
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Two angles and the included side are equal. Angle-Side-Angle: side must be between the two angles.
Two angles and the included side are equal. Angle-Side-Angle: side must be between the two angles.
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What does the similarity statement $\triangle ABC \sim \triangle DEF$ tell you about correspondence?
What does the similarity statement $\triangle ABC \sim \triangle DEF$ tell you about correspondence?
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$A\leftrightarrow D$, $B\leftrightarrow E$, $C\leftrightarrow F$. Order of vertices shows which parts correspond.
$A\leftrightarrow D$, $B\leftrightarrow E$, $C\leftrightarrow F$. Order of vertices shows which parts correspond.
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Which transformation changes size but preserves shape and angles?
Which transformation changes size but preserves shape and angles?
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A dilation. Dilations change size while preserving shape.
A dilation. Dilations change size while preserving shape.
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Identify the missing value: if similar triangles have $k=\frac{3}{2}$ and a side is $10$, what is the corresponding side?
Identify the missing value: if similar triangles have $k=\frac{3}{2}$ and a side is $10$, what is the corresponding side?
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$15$. Multiply original side by scale factor: $10 \times \frac{3}{2} = 15$.
$15$. Multiply original side by scale factor: $10 \times \frac{3}{2} = 15$.
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Identify whether $SSS$ similarity applies: sides are $3,4,5$ and $6,8,10$; are the triangles similar?
Identify whether $SSS$ similarity applies: sides are $3,4,5$ and $6,8,10$; are the triangles similar?
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Yes, by $SSS$ similarity with $k=2$. All sides proportional with ratio $\frac{6}{3} = 2$.
Yes, by $SSS$ similarity with $k=2$. All sides proportional with ratio $\frac{6}{3} = 2$.
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Identify whether $AA$ similarity applies: one triangle has angles $40^\circ,60^\circ,80^\circ$ and another has $40^\circ,60^\circ,80^\circ$.
Identify whether $AA$ similarity applies: one triangle has angles $40^\circ,60^\circ,80^\circ$ and another has $40^\circ,60^\circ,80^\circ$.
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Yes, by $AA$ similarity. Two pairs of angles are equal.
Yes, by $AA$ similarity. Two pairs of angles are equal.
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Find $x$: if similar triangles satisfy $\frac{AB}{DE}=\frac{4}{7}$ and $DE=21$, what is $AB$?
Find $x$: if similar triangles satisfy $\frac{AB}{DE}=\frac{4}{7}$ and $DE=21$, what is $AB$?
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$12$. Cross multiply: $AB = DE \times \frac{4}{7} = 21 \times \frac{4}{7} = 12$.
$12$. Cross multiply: $AB = DE \times \frac{4}{7} = 21 \times \frac{4}{7} = 12$.
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Find the new perimeter: a figure has perimeter $28$ and is dilated by $k=\frac{5}{2}$; what is the new perimeter?
Find the new perimeter: a figure has perimeter $28$ and is dilated by $k=\frac{5}{2}$; what is the new perimeter?
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$70$. New perimeter is $28 \times \frac{5}{2} = 70$.
$70$. New perimeter is $28 \times \frac{5}{2} = 70$.
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What triangle congruence criterion is abbreviated $SAS$?
What triangle congruence criterion is abbreviated $SAS$?
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Two sides and the included angle are equal. Side-Angle-Side: angle must be between the two sides.
Two sides and the included angle are equal. Side-Angle-Side: angle must be between the two sides.
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What does it mean for two figures to be congruent?
What does it mean for two figures to be congruent?
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Same shape and size; all corresponding sides and angles are equal. Congruent figures are identical copies.
Same shape and size; all corresponding sides and angles are equal. Congruent figures are identical copies.
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What is the relationship between corresponding angles in similar triangles?
What is the relationship between corresponding angles in similar triangles?
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Corresponding angles are equal in measure. Similar triangles preserve all angle measures.
Corresponding angles are equal in measure. Similar triangles preserve all angle measures.
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What is the area relationship for similar figures with scale factor $k$?
What is the area relationship for similar figures with scale factor $k$?
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Areas scale by $k^2$. Area scales by the square of the linear factor.
Areas scale by $k^2$. Area scales by the square of the linear factor.
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Identify the scale factor: a side changes from $12$ to $18$ under similarity; what is $k$?
Identify the scale factor: a side changes from $12$ to $18$ under similarity; what is $k$?
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$\frac{3}{2}$. Divide new length by original: $\frac{18}{12} = \frac{3}{2}$.
$\frac{3}{2}$. Divide new length by original: $\frac{18}{12} = \frac{3}{2}$.
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Find the missing angle: if $\triangle ABC \sim \triangle DEF$ and $\angle A=35^\circ$, what is $\angle D$?
Find the missing angle: if $\triangle ABC \sim \triangle DEF$ and $\angle A=35^\circ$, what is $\angle D$?
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$35^\circ$. Corresponding angles in similar triangles are equal.
$35^\circ$. Corresponding angles in similar triangles are equal.
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How does congruence differ from similarity?
How does congruence differ from similarity?
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Congruence requires equal sides and angles; similarity requires proportional sides and equal angles. Similarity allows scaling; congruence requires exact match.
Congruence requires equal sides and angles; similarity requires proportional sides and equal angles. Similarity allows scaling; congruence requires exact match.
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If two triangles are similar, what can be said about their perimeters?
If two triangles are similar, what can be said about their perimeters?
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Perimeters are in the same ratio as their corresponding sides. Perimeter scales with linear dimensions.
Perimeters are in the same ratio as their corresponding sides. Perimeter scales with linear dimensions.
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What is the definition of similar triangles?
What is the definition of similar triangles?
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Triangles with corresponding angles equal and sides proportional. This defines similarity: shape preserved, size scaled.
Triangles with corresponding angles equal and sides proportional. This defines similarity: shape preserved, size scaled.
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Which theorem states triangles are congruent if two sides and an included angle are equal?
Which theorem states triangles are congruent if two sides and an included angle are equal?
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Side-Angle-Side (SAS) congruence theorem. Two sides and the angle between them.
Side-Angle-Side (SAS) congruence theorem. Two sides and the angle between them.
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Determine if triangles with sides 3, 4, 5 and 6, 8, 10 are similar.
Determine if triangles with sides 3, 4, 5 and 6, 8, 10 are similar.
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Yes, they are similar by SSS similarity theorem. Sides are in ratio $2:1$, proving similarity.
Yes, they are similar by SSS similarity theorem. Sides are in ratio $2:1$, proving similarity.
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Can right triangles be similar to non-right triangles?
Can right triangles be similar to non-right triangles?
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No, a right triangle cannot be similar to a non-right triangle. Different angle sets prevent similarity.
No, a right triangle cannot be similar to a non-right triangle. Different angle sets prevent similarity.
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Given similar triangles, what is the ratio of their circumferences?
Given similar triangles, what is the ratio of their circumferences?
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The ratio of circumferences equals the ratio of corresponding sides. Linear measurements scale proportionally.
The ratio of circumferences equals the ratio of corresponding sides. Linear measurements scale proportionally.
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Find the missing side of a similar triangle with sides 3, 4, and ? given similar triangle sides 6, 8, 10.
Find the missing side of a similar triangle with sides 3, 4, and ? given similar triangle sides 6, 8, 10.
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The missing side is 5. Use proportional sides: $\frac{3}{6} = \frac{?}{10}$.
The missing side is 5. Use proportional sides: $\frac{3}{6} = \frac{?}{10}$.
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What is the effect of similarity on the areas of two triangles?
What is the effect of similarity on the areas of two triangles?
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The ratio of the areas is the square of the ratio of corresponding sides. Area scales with the square of linear scale factor.
The ratio of the areas is the square of the ratio of corresponding sides. Area scales with the square of linear scale factor.
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Are two isosceles triangles always similar?
Are two isosceles triangles always similar?
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No, isosceles triangles are not always similar. Different base angles prevent guaranteed similarity.
No, isosceles triangles are not always similar. Different base angles prevent guaranteed similarity.
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What is the Angle-Side-Angle (ASA) congruence postulate?
What is the Angle-Side-Angle (ASA) congruence postulate?
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Triangles are congruent if two angles and the included side are equal. Two angles and the side between them.
Triangles are congruent if two angles and the included side are equal. Two angles and the side between them.
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