Triangle Similarity Theorems and Pythagorean Theorem - Geometry

1. If $\triangle ABC \sim \triangle DEF$ and $\frac{AB}{DE}=\frac{2}{5}$, what is $k$ from $ABC$ to $DEF$?

   $k=\frac{5}{2}$. Scale factor is the reciprocal: $k = \frac{1}{\frac{2}{5}} = \frac{5}{2}$.

2. What is the third angle if a triangle has angles $28^\circ$ and $92^\circ$?

   $60^\circ$. Third angle: $180^\circ - 28^\circ - 92^\circ = 60^\circ$.

3. If $\angle A \cong \angle D$ and $\angle B \cong \angle E$, what similarity statement matches the order?

   $\triangle ABC \sim \triangle DEF$. The order of vertices shows the correspondence.

4. What is the key conclusion of AA similarity about side lengths?

   Corresponding sides are proportional. Similar triangles have proportional corresponding sides.

5. What does it mean for triangles to be similar in terms of a transformation?

   One can be mapped to the other by a similarity transformation. This is the transformation definition of similarity.

6. What is the correct proportionality statement if $\triangle ABC \sim \triangle DEF$?

   $\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$. This shows the correct order of corresponding sides.

7. Choose the correct correspondence: $\triangle ABC \sim \triangle RST$. Which side matches $\overline{BC}$?

   $\overline{ST}$. Second and third sides correspond by position.

8. In $\triangle ABC \sim \triangle DEF$, which side corresponds to $\overline{AC}$?

   $\overline{DF}$. Corresponding sides match by position in the similarity statement.

9. Identify the correct proportionality if $\triangle ABC \sim \triangle DEF$ and the order gives $A\leftrightarrow D$.

   $\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$. The correspondence order determines the proportionality.

10. If two triangles have two pairs of equal angles, what must be true about their angle sets?

    All three pairs of corresponding angles are congruent. If two pairs match, the third pair also matches by angle sum.

11. What is true about corresponding side lengths under a dilation of factor $k$?

    Each corresponding length is multiplied by $k$. Dilation scales all distances by the same factor.

12. What is the minimum angle information needed to prove two triangles similar by AA?

    Two pairs of corresponding congruent angles. AA requires exactly two pairs of congruent corresponding angles.

13. If $m\angle D=50^\circ$ and $m\angle E=60^\circ$, what is $m\angle F$?

    $70^\circ$. Angles sum to $180^\circ$: $180^\circ - 50^\circ - 60^\circ = 70^\circ$.

14. If $\triangle ABC \sim \triangle DEF$ and $AB=8$, $DE=12$, what is the scale factor from $ABC$ to $DEF$?

    $k=\frac{12}{8}=\frac{3}{2}$. Scale factor is the ratio of corresponding sides.

15. If two triangles are similar by AA, what is the relationship between their perimeters?

    Perimeters are in the same ratio as the scale factor $k$. Linear dimensions scale by $k$, so perimeters scale by $k$.

16. Find $EF$ if $\triangle ABC \sim \triangle DEF$, $BC=9$, and $\frac{BC}{EF}=\frac{3}{5}$.

    $EF=15$. Cross-multiply: $9 \times 5 = 3 \times EF$, so $EF = 15$.

17. What is the scale factor from $\triangle DEF$ to $\triangle ABC$ if $DE=10$ corresponds to $AB=4$?

    $k=\frac{4}{10}=\frac{2}{5}$. Scale factor is the ratio from first to second triangle.

18. If $\triangle ABC \sim \triangle DEF$ and $k=2$, what is $DE$ when $AB=7$?

    $DE=14$. Multiply corresponding side by scale factor: $7 \times 2 = 14$.

19. If $\triangle ABC \sim \triangle DEF$ and $AC=15$, $DF=20$, what is $k$ from $ABC$ to $DEF$?

    $k=\frac{20}{15}=\frac{4}{3}$. Scale factor is the ratio of corresponding sides.

20. Which option best justifies AA using transformations after aligning one angle and a side?

    A dilation about the vertex maps the second side to the corresponding side. The dilation maps the aligned triangle to match size.

21. What does it mean for two angles to be congruent?

    They have equal measure. Congruent angles have identical measures.

22. What is preserved by a similarity transformation: angles, ratios of lengths, or both?

    Both angle measures and ratios of corresponding lengths. Similarity transformations preserve both angles and length ratios.

23. If $\triangle ABC \sim \triangle DEF$ and $k=\frac{1}{3}$ from $ABC$ to $DEF$, what is $EF$ when $BC=12$?

    $EF=4$. Multiply corresponding side by scale factor: $12 \times \frac{1}{3} = 4$.

24. What is the role of a dilation in an AA similarity proof using transformations?

    It matches the size after angles are aligned by rigid motions. Dilation scales the triangle to match the second triangle's size.

25. Choose the correct correspondence: $\triangle ABC \sim \triangle RST$. Which angle matches $\angle B$?

    $\angle S$. Second angle corresponds to second angle in the order.

26. Find and correct the statement: "AA similarity requires three pairs of congruent angles."

    Correct: AA similarity requires two pairs of congruent angles. AA only requires two pairs, not all three.

27. What is the scale factor of a dilation if every length is multiplied by $k$?

    The scale factor is $k$. The factor $k$ multiplies all corresponding lengths.

28. If $\triangle ABC \sim \triangle DEF$ with scale factor $k$, what is $\frac{AB}{DE}$ in terms of $k$?

    $\frac{AB}{DE}=\frac{1}{k}$ (from $ABC$ to $DEF$). The ratio is the reciprocal of the scale factor.

29. Identify whether AA applies: one triangle has angles $40^\circ$ and $70^\circ$, the other has $40^\circ$ and $70^\circ$.

    Yes, the triangles are similar by AA. Two pairs of equal angles satisfy the AA criterion.

30. Which transformations are included in a similarity transformation: translation, rotation, reflection, dilation?

    All of them (rigid motions plus dilation). Similarity transformations include all isometries plus dilation.

31. Find and correct the statement: "Rigid motions change distances by a scale factor."

    Correct: rigid motions preserve distances exactly. Rigid motions preserve all distances exactly.

32. What is the key conclusion of AA similarity about angle measures?

    All corresponding angles are congruent. Similar triangles have all corresponding angles congruent.

33. Identify the missing step: rigid motion maps one vertex and one side; what is done next to match size?

    Apply a dilation with scale factor $k$. Dilation adjusts size after rigid motions align position.

34. If $\angle A \cong \angle D$ and $\angle B \cong \angle E$, which angles must also be congruent?

    $\angle C \cong \angle F$. The third pair must be congruent by the angle sum theorem.

35. What is the Triangle Angle Sum Theorem used with AA similarity?

    In any triangle, $m\angle A+m\angle B+m\angle C=180^\circ$. The angle sum is always $180^\circ$ in any triangle.

36. What is the relationship between corresponding side ratios in similar triangles?

    All corresponding side ratios are equal to the same constant $k$. The constant ratio $k$ is the scale factor.

37. Which statement is always true if triangles are similar by AA?

    There exists a dilation and rigid motions mapping one triangle to the other. This transformation characterizes similar triangles.

38. If $m\angle A=50^\circ$, $m\angle B=60^\circ$, what is $m\angle C$?

    $70^\circ$. Angles sum to $180^\circ$: $180^\circ - 50^\circ - 60^\circ = 70^\circ$.

39. Find the missing ratio if $\triangle ABC \sim \triangle DEF$ and $\frac{AB}{DE}=\frac{3}{4}$: what is $\frac{BC}{EF}$?

    $\frac{3}{4}$. All corresponding ratios are equal in similar triangles.

40. What is the meaning of $\triangle ABC \sim \triangle DEF$?

    Corresponding angles are congruent and corresponding sides are proportional. The symbol $\sim$ denotes similarity with these two properties.

41. In $\triangle ABC \sim \triangle DEF$, which angle corresponds to $\angle B$?

    $\angle E$. Corresponding angles match by position in the similarity statement.

42. If $\triangle ABC \sim \triangle DEF$ and $k=\frac{3}{4}$ from $ABC$ to $DEF$, what is $AB$ when $DE=21$?

    $AB=28$. Use inverse scale factor: $AB = \frac{21}{\frac{3}{4}} = 21 \times \frac{4}{3} = 28$.

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

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

44. If two angles of one triangle match two angles of another, what can you conclude about the third angles?

    The third angles are congruent. The angle sum theorem ensures the third angles are equal.

45. What is true about corresponding angles under a dilation?

    Corresponding angles remain congruent. Dilations preserve angle measures exactly.

46. Find and correct the statement: "A dilation changes angle measures."

    Correct: a dilation preserves angle measures. Dilations preserve angles, changing only size.

47. Which option proves similarity: $\angle A \cong \angle D$ and $\angle C \cong \angle F$?

    AA similarity applies, so the triangles are similar. Two pairs of congruent angles prove similarity by AA.

48. Identify whether AA applies: one triangle has angles $35^\circ$ and $80^\circ$, the other has $35^\circ$ and $75^\circ$.

    No, AA does not apply. Different angle measures mean the triangles are not similar.

49. What property of rigid motions is used in proving AA similarity?

    Rigid motions preserve angle measure and distance. These properties ensure angles remain unchanged during transformation.

50. If $\triangle ABC \sim \triangle DEF$ and $AB:DE=5:2$, what is $k$ from $DEF$ to $ABC$?

    $k=\frac{5}{2}$. The ratio $AB:DE = 5:2$ means scale factor from $DEF$ to $ABC$ is $\frac{5}{2}$.

51. What property of dilations is used in proving AA similarity?

    Dilations preserve angle measure and scale all lengths by $k$. Angles stay the same while sides scale uniformly.

52. Identify the conclusion: $\angle A \cong \angle D$ and $\angle B \cong \angle E$ for two triangles.

    The triangles are similar by AA. Two pairs of congruent angles satisfy the AA criterion.

53. What is a similarity transformation in the plane?

    A composition of rigid motions and a dilation. This defines the complete similarity transformation.

54. Find $AC$ if $\triangle ABC \sim \triangle DEF$, $DF=18$, and $k=\frac{3}{2}$ from $ABC$ to $DEF$.

    $AC=12$. Use inverse scale factor: $AC = \frac{18}{\frac{3}{2}} = 18 \times \frac{2}{3} = 12$.

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