Trigonometric Ratios from Right Triangle Similarity - Geometry

1. What is the scale factor from triangle $\triangle ABC$ to $\triangle DEF$ if $AB=6$ and $DE=9$ are corresponding?

   $k=\frac{9}{6}=\frac{3}{2}$. Scale factor is the ratio of corresponding sides.

2. Which statement is always true if $\triangle ABC\cong\triangle DEF$: $\frac{AB}{DE}=1$ or $\frac{AB}{DE}=k$ for any $k$?

   $\frac{AB}{DE}=1$. Congruent triangles have scale factor 1 (equal sides).

3. What does it mean for two triangles to be congruent in terms of corresponding parts?

   $\text{All corresponding sides and angles are equal}$. Congruent triangles are identical in size and shape.

4. Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=9$, $DE=6$, and $AC=x$ corresponds to $DF=10$.

   $x=10\cdot\frac{9}{6}=15$. Scale factor from ABC to DEF is $\frac{9}{6} = \frac{3}{2}$.

5. Find $BC$ if $\triangle ABC\sim\triangle DEF$, $k=\frac{3}{2}$ from $ABC$ to $DEF$, and $EF=12$ corresponds to $BC$.

   $BC=12\cdot\frac{2}{3}=8$. Reverse the scale factor: $BC = EF \cdot \frac{2}{3}$.

6. What similarity criterion states that if two angles are congruent, then the triangles are similar?

   $\text{AA similarity}$. Two angles determine the third, making triangles similar.

7. Identify the criterion: $\frac{AB}{DE}=\frac{BC}{EF}$ and $\angle B\cong\angle E$ where the angle is included.

   $\text{SAS similarity}$. Two proportional sides with included angle congruent.

8. Identify the criterion: $\angle A\cong\angle D$, $\angle B\cong\angle E$, and $AC\cong DF$ where the side is not included.

   $\text{AAS congruence}$. Two angles and non-included side match the AAS pattern.

9. Which triangle criterion proves similarity when two sides are in ratio $\frac{5}{8}$ and $\frac{10}{16}$ and the included angle is equal?

   $\text{SAS similarity}$. Ratios $\frac{5}{8} = \frac{10}{16}$ with included angle match SAS.

10. What congruence criterion states that if two sides and the included angle are equal, then the triangles are congruent?

    $\text{SAS congruence}$. Two sides with their included angle determine triangle uniquely.

11. What does it mean for sides to be proportional in similar triangles using a scale factor $k$?

    $\text{Each corresponding side ratio equals }k$. Each ratio $\frac{\text{side}_1}{\text{side}_2} = k$ for all pairs.

12. If $\triangle ABC\sim\triangle DEF$, which side corresponds to $AB$?

    $DE$. First sides in similarity statement correspond.

13. What must be true about the angle in $\text{SAS}$ similarity to use the criterion correctly?

    $\text{It must be the included angle between the proportional sides}$. The angle must be between the two proportional sides.

14. Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=8$, $DE=12$, and $BC=x$ corresponds to $EF=15$.

    $x=\frac{8}{12}\cdot 15=10$. Use proportion: $\frac{8}{12} = \frac{x}{15}$.

15. Identify the criterion: In right triangles, $\angle C=\angle F=90^\circ$, $AC=DF$, and $BC=EF$.

    $\text{SAS congruence}$. Two sides and included right angle match SAS.

16. If $\triangle ABC\sim\triangle DEF$, which angle corresponds to $\angle C$?

    $\angle F$. Third angles in similarity statement correspond.

17. Find $x$ if $\triangle ABC\sim\triangle DEF$, $\frac{AB}{DE}=\frac{3}{7}$, and $AB=12$ with $DE=x$.

    $x=12\cdot\frac{7}{3}=28$. If $\frac{AB}{DE} = \frac{3}{7}$, then $DE = AB \cdot \frac{7}{3}$.

18. What must be true about the side in $\text{ASA}$ congruence to use the criterion correctly?

    $\text{It must be the included side between the two angles}$. The side must be between the two congruent angles.

19. What similarity criterion states that if two pairs of sides are proportional and the included angle is congruent, then triangles are similar?

    $\text{SAS similarity}$. Two proportional sides with included angle congruent.

20. What does it mean for two triangles to be similar in terms of angles and side ratios?

    $\text{Corresponding angles congruent; corresponding sides proportional}$. Similar triangles have matching angles and proportional sides.

21. What congruence criterion states that if two angles and a non-included side are equal, then the triangles are congruent?

    $\text{AAS congruence}$. Two angles determine the third, making triangles congruent.

22. Which congruence criterion requires that the triangles are right triangles: $\text{HL}$ or $\text{AAS}$?

    $\text{HL}$. HL applies only to right triangles.

23. Identify the missing reason: If $\triangle ABC\cong\triangle DEF$, then $\angle B\cong\angle E$ by what principle?

    $\text{CPCTC}$. Corresponding Parts of Congruent Triangles are Congruent.

24. What is the similarity statement if $\angle A\cong\angle D$, $\angle B\cong\angle E$, and $\angle C\cong\angle F$?

    $\triangle ABC\sim\triangle DEF$. Corresponding vertices are written in matching order.

25. Which congruence criterion is invalid in general and may produce two different triangles: $\text{SSA}$ or $\text{SAS}$?

    $\text{SSA}$. SSA can create ambiguous cases with two possible triangles.

26. Find $x$ if similar triangles have scale factor (small to large) $k=\frac{5}{2}$ and the small side is $x$ while the large is $20$.

    $x=20\cdot\frac{2}{5}=8$. Small triangle side times $\frac{2}{5}$ equals large side.

27. What is the relationship between the scale factor $k$ of similar triangles and their perimeters?

    $\frac{P_1}{P_2}=k$. Perimeters scale linearly with the scale factor.

28. Identify the criterion: Two right triangles have equal hypotenuse and one equal leg.

    $\text{HL congruence}$. Hypotenuse-Leg is specific to right triangles.

29. What right-triangle congruence criterion uses hypotenuse and one leg to prove triangles congruent?

    $\text{HL congruence}$. Hypotenuse-Leg theorem for right triangles only.

30. Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=12$ corresponds to $DE=9$, and $\text{perimeter}(ABC)=40$ with $\text{perimeter}(DEF)=x$.

    $x=40\cdot\frac{9}{12}=30$. Perimeters scale with same factor as sides: $\frac{9}{12}$.

31. Identify the criterion: $\angle A\cong\angle D$ and $\angle C\cong\angle F$ for two triangles.

    $\text{AA similarity}$. Two congruent angles establish AA similarity.

32. Identify the criterion: $\angle A\cong\angle D$, $\angle B\cong\angle E$, and $AB\cong DE$ where the side is included.

    $\text{ASA congruence}$. Two angles and included side match the ASA pattern.

33. Find the ratio of perimeters if two similar triangles have scale factor $k=\frac{4}{3}$ from triangle $1$ to triangle $2$.

    $\frac{P_2}{P_1}=\frac{4}{3}$. Perimeter ratio equals the scale factor directly.

34. Identify the criterion: $AB=DE$, $BC=EF$, and $\angle B\cong\angle E$ where the angle is included between the given sides.

    $\text{SAS congruence}$. Two sides and included angle match the SAS pattern.

35. What does $\text{CPCTC}$ stand for in triangle congruence proofs?

    $\text{Corresponding parts of congruent triangles are congruent}$. Standard definition of the CPCTC theorem.

36. Find $EF$ if $\triangle ABC\sim\triangle DEF$, $k=\frac{3}{2}$ from $ABC$ to $DEF$, and $BC=8$ corresponds to $EF$.

    $EF=8\cdot\frac{3}{2}=12$. Apply scale factor: $EF = BC \cdot k$.

37. What conclusion follows if $\triangle ABC\sim\triangle DEF$ regarding $\frac{AB}{DE}$ and $\frac{BC}{EF}$?

    $\frac{AB}{DE}=\frac{BC}{EF}$. Similar triangles have all corresponding side ratios equal.

38. What congruence criterion states that if three pairs of sides are equal, then the triangles are congruent?

    $\text{SSS congruence}$. Three sides determine triangle shape uniquely.

39. Find the scale factor $k$ (from $\triangle ABC$ to $\triangle DEF$) if $BC=11$ corresponds to $EF=22$.

    $k=\frac{22}{11}=2$. Scale factor is ratio of corresponding sides.

40. Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=6$ corresponds to $DE=15$, and $\text{area}(ABC)=24$ with $\text{area}(DEF)=x$.

    $x=24\cdot\left(\frac{15}{6}\right)^2=150$. Areas scale with square of side ratio: $\left(\frac{15}{6}\right)^2$.

41. Which criterion proves similarity when you know only that $\angle A\cong\angle D$ and $\angle B\cong\angle E$?

    $\text{AA similarity}$. Two congruent angles are sufficient for AA similarity.

42. Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=5$, $AC=7$, $DE=15$, and $DF=x$ corresponds to $AC$.

    $x=7\cdot\frac{15}{5}=21$. Scale factor from ABC to DEF is $\frac{15}{5} = 3$.

43. Find the ratio of areas if two similar triangles have scale factor $k=3$ from the smaller to the larger.

    $\frac{A_{\text{large}}}{A_{\text{small}}}=9$. Area ratio equals scale factor squared: $3^2 = 9$.

44. What is the relationship between the scale factor $k$ of similar triangles and their areas?

    $\frac{A_1}{A_2}=k^2$. Areas scale with the square of the scale factor.

45. What conclusion follows if $\triangle ABC\sim\triangle DEF$ regarding $\angle A$ and $\angle D$?

    $\angle A\cong\angle D$. Similar triangles have all corresponding angles congruent.

46. What congruence criterion states that if two angles and the included side are equal, then the triangles are congruent?

    $\text{ASA congruence}$. Two angles with their included side determine triangle uniquely.

47. Which criterion guarantees triangle congruence when $\angle A\cong\angle D$, $\angle B\cong\angle E$, and $\angle C\cong\angle F$?

    $\text{None; AAA gives similarity, not congruence}$. AAA proves similarity only, not congruence.

48. Find $x$ if $\triangle ABC\sim\triangle DEF$, $\frac{AB}{DE}=\frac{2}{5}$, and $DE=20$ with $AB=x$.

    $x=\frac{2}{5}\cdot 20=8$. Given ratio tells us $AB = \frac{2}{5} \cdot DE$.

49. Identify the criterion: $\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$ for two triangles.

    $\text{SSS similarity}$. All three side ratios are equal in SSS similarity.

50. Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=4$, $BC=6$, $DE=10$, and $EF=x$ corresponds to $BC$.

    $x=6\cdot\frac{10}{4}=15$. Scale factor from ABC to DEF is $\frac{10}{4} = \frac{5}{2}$.

51. What similarity criterion states that if three pairs of corresponding sides are proportional, then triangles are similar?

    $\text{SSS similarity}$. All three side ratios equal means triangles are similar.

52. Find $x$ if $\triangle ABC\sim\triangle DEF$, $AC=14$ corresponds to $DF=21$, and $BC=10$ corresponds to $EF=x$.

    $x=10\cdot\frac{21}{14}=15$. Scale factor is $\frac{21}{14} = \frac{3}{2}$.

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