Congruence and Similarity - ISEE Upper Level Quantitative Reasoning

1. Which condition is NOT sufficient for triangle congruence: $AAA$, $SSS$, $SAS$, or $ASA$?

   $AAA$. AAA only ensures similarity by matching angles but does not guarantee equal side lengths for congruence.

2. What is a rigid motion that preserves congruence (list the allowed transformations)?

   Translation, rotation, reflection (and their combinations). Rigid motions preserve distances and angles, maintaining congruence without altering size or shape.

3. What is the scale factor from a figure to its image if each length is multiplied by $k$?

   Scale factor $=k$. The scale factor $k$ represents the constant multiplier applied to all linear dimensions in a similarity transformation.

4. Identify the conclusion: if $\triangle ABC \cong \triangle DEF$, what is the value of $\angle B-\angle E$?

   $0^\circ$. Congruent triangles have equal corresponding angles, so $\angle B = \angle E$, making their difference $0^\circ$.

5. What is the triangle similarity criterion $SAS$ (similarity version)?

   Included angle equal and adjacent sides proportional. SAS similarity requires proportional sides adjacent to an equal included angle, preserving shape proportionally.

6. Identify the missing side: if $\triangle ABC \sim \triangle DEF$, $\frac{AB}{DE}=\frac{3}{5}$, and $AB=12$, what is $DE$?

   $DE=20$. Cross-multiplying the proportion $\frac{AB}{DE}=\frac{3}{5}$ with $AB=12$ yields $DE=20$ due to similarity ratios.

7. Find $x$: if $\triangle ABC \sim \triangle DEF$ and the scale factor is $k=4$ (image to original), what is $EF$ when $BC=7$?

   $EF=28$. With scale factor $k=4$ (image to original), $EF=BC \times 4=7 \times 4=28$ due to proportional sides.

8. What is the right-triangle congruence criterion $HL$?

   Hypotenuse and one leg are equal in right triangles. HL congruence for right triangles requires the hypotenuse and one leg to match, sufficient due to the right angle.

9. If two similar rectangles have scale factor $k=\frac{3}{2}$, what is the area ratio (image to original)?

   $\frac{9}{4}$. The area ratio for similar figures is the square of the scale factor, yielding $\left(\frac{3}{2}\right)^2=\frac{9}{4}$.

10. If two similar solids have volume ratio (image to original) $\frac{27}{8}$, what is the linear scale factor $k$?

    $k=\frac{3}{2}$. The scale factor $k$ is the cube root of the volume ratio, so $k=\sqrt[3]{\frac{27}{8}}=\frac{3}{2}$.

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

    Same shape and size; all corresponding sides and angles are equal. Congruence requires identical shapes and sizes, ensuring all corresponding parts match exactly.

12. If similar figures have scale factor $k$, what is the ratio of their perimeters (image to original)?

    Perimeter ratio $=k$. Perimeters of similar figures scale linearly with the scale factor $k$, as they are sums of proportional sides.

13. If similar figures have scale factor $k$, what is the ratio of their areas (image to original)?

    Area ratio $=k^2$. Areas scale with the square of the linear scale factor $k$ due to two-dimensional proportionality.

14. What is the triangle similarity criterion $AA$?

    Two pairs of corresponding angles are equal. AA similarity is proven by two equal corresponding angles, implying the third angle also equals by triangle sum.

15. What type of transformation maps a figure to a similar figure with scale factor $k \ne 1$?

    Dilation. Dilation enlarges or reduces figures by a scale factor $k \ne 1$, preserving angles and proportionality for similarity.

16. What is the triangle similarity criterion $SSS$ (similarity version)?

    All $3$ pairs of corresponding sides are proportional. SSS similarity holds when all corresponding sides are proportional, ensuring equal angles via scaling.

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

    Same shape; corresponding angles equal and sides proportional. Similarity preserves shape through equal corresponding angles and proportional sides, allowing different sizes.

18. What is the triangle congruence criterion $SSS$?

    All $3$ pairs of corresponding sides are equal. SSS congruence holds when all three sides of one triangle equal the corresponding sides of another, proving identical shapes.

19. Find $x$: if $\triangle ABC \sim \triangle DEF$ and $AB=10$, $DE=15$, $AC=12$, what is $DF$?

    $DF=18$. The scale factor $\frac{DE}{AB}=\frac{15}{10}=\frac{3}{2}$ applies to $AC$, giving $DF=12 \times \frac{3}{2}=18$.

20. Find the scale factor (image to original): a side changes from $8$ to $14$ under a dilation. What is $k$?

    $k=\frac{7}{4}$. The scale factor $k$ is the ratio of new length to original, so $k=\frac{14}{8}=\frac{7}{4}$ under dilation.

21. What is the triangle congruence criterion $SAS$?

    Two sides and the included angle are equal. SAS congruence applies when two sides and the angle between them match in both triangles, determining the third side.

22. If similar solids have scale factor $k$, what is the ratio of their volumes (image to original)?

    Volume ratio $=k^3$. Volumes of similar solids scale with the cube of the linear scale factor $k$ because of three-dimensional expansion.

23. What is the triangle congruence criterion $AAS$?

    Two angles and a non-included side are equal. AAS congruence occurs with two matching angles and a non-adjacent side, implying the third angle also matches.

24. Find $x$: if $\triangle ABC \sim \triangle DEF$, $\frac{BC}{EF}=\frac{2}{3}$, $EF=18$, what is $BC$?

    $BC=12$. Solving the proportion $\frac{BC}{EF}=\frac{2}{3}$ with $EF=18$ gives $BC=12$ from similar triangle ratios.

25. What is the triangle congruence criterion $ASA$?

    Two angles and the included side are equal. ASA congruence is established by matching two angles and the side between them, fixing the remaining elements.

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