Theorems about Lines and Angles - Geometry

1. In ASA, if $\overline{AB}$ is fixed and both endpoint angles are fixed, what is fixed about the third vertex?

   It is the intersection of the two determined rays. Two rays from fixed points have unique intersection.

2. What does SSS stand for in triangle congruence criteria?

   Side-Side-Side. All three pairs of corresponding sides are equal.

3. What does ASA stand for in triangle congruence criteria?

   Angle-Side-Angle. Two angles and the side between them are equal.

4. What does it mean for two figures to be congruent in terms of rigid motions?

   One can be mapped to the other by rigid motions. Congruent means same shape and size.

5. Which statement best matches ASA: $\angle A = \angle D$, $AB = DE$, $\angle B = \angle E$ or $AB = DE$, $BC = EF$, $AC = DF$?

   $\angle A = \angle D$, $AB = DE$, $\angle B = \angle E$. Two angles with their included side are given.

6. What is the key rigid-motion idea behind proving SAS congruence?

   Match a side and included angle; the second side fixes the third vertex. The angle direction constrains the third vertex location.

7. Choose the correct included side for angles $\angle B$ and $\angle C$ in $\triangle ABC$: $\overline{AB}$, $\overline{BC}$, or $\overline{AC}$.

   $\overline{BC}$. Side $BC$ connects angles $\angle B$ and $\angle C$.

8. What is the congruence criterion if $\angle A = \angle D$, $AC = DF$, and $\angle C = \angle F$?

   ASA (the included side is $\overline{AC}$). Side $AC$ is between the two given angles.

9. What property of segment length is preserved by any rigid motion?

   If $AB = CD$, then $A'B' = C'D'$. Distances remain unchanged under rigid motions.

10. Choose the correct included angle for sides $AB$ and $BC$ in $\triangle ABC$: $\angle A$, $\angle B$, or $\angle C$.

    $\angle B$. Vertex $B$ is where sides $AB$ and $BC$ meet.

11. Which congruence criterion uses three pairs of corresponding sides?

    SSS. Uses all three pairs of corresponding sides.

12. Which side is required in ASA: the included side or a non-included side?

    The included side between the two angles. The side must be between the two given angles.

13. In SAS, if two triangles share the same included angle and adjacent side lengths, what is fixed about the third vertex position?

    It is uniquely determined up to reflection. Only reflection can change orientation.

14. Which criterion is invalid: SSA, SAS, or SSS?

    SSA is not a valid congruence criterion in general. SSA allows ambiguous cases with two possible triangles.

15. Which fact ensures a triangle is determined by three side lengths (when possible)?

    The third vertex is at the intersection of two circles. Distance constraints create unique intersection points.

16. What is the key rigid-motion idea behind proving SSS congruence?

    Match one side, then the third vertex is fixed by two distances. Two circle intersections determine the third vertex.

17. What does it mean for two triangles to be congruent using vertex correspondence?

    $\triangle ABC \cong \triangle DEF$ means $A\leftrightarrow D$, $B\leftrightarrow E$, $C\leftrightarrow F$. Each vertex maps to its corresponding vertex.

18. Which statement best matches SAS: $AB = DE$, $\angle A = \angle D$, $AC = DF$ or $\angle A = \angle D$, $AB = DE$, $\angle B = \angle E$?

    $AB = DE$, $\angle A = \angle D$, $AC = DF$. Two sides with their included angle are given.

19. What is the key rigid-motion idea behind proving ASA congruence?

    Match the included side; the two angles fix the rays to the third vertex. Two angle directions intersect at the third vertex.

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

    $\overline{DF}$. Vertices correspond in the same order: $C \leftrightarrow F$.

21. In ASA, if $\angle A = \angle D$, $AB = DE$, and $\angle B = \angle E$, what is the conclusion?

    $\triangle ABC \cong \triangle DEF$ by ASA. Two angles and included side establish congruence.

22. In $\triangle ABC \cong \triangle DEF$, which angle corresponds to $\angle B$?

    $\angle E$. Vertices correspond in the same order: $B \leftrightarrow E$.

23. What rigid-motion fact justifies concluding corresponding sides are equal after mapping triangles?

    Rigid motions preserve distance between all pairs of points. All corresponding distances are preserved.

24. Identify the issue: $\angle A = \angle D$, $\angle C = \angle F$, and $AB = DE$ claimed as ASA.

    The side is not included; this is AAS, not ASA. ASA requires the side between the two given angles.

25. Identify the correct conclusion: If a rigid motion maps $\triangle ABC$ onto $\triangle DEF$, then $AB$ and $DE$ are what?

    Congruent segments, so $AB = DE$. Rigid motions preserve all distances between points.

26. What three transformations are typically used as rigid motions in Geometry?

    Translations, rotations, and reflections. These preserve distances and angles without distortion.

27. What is the rigid-motion reason that SSS can produce at most two mirror-image triangles?

    A reflection can swap the two circle-intersection positions. Two circles intersect at most at two points.

28. What rigid-motion fact justifies concluding corresponding angles are equal after mapping triangles?

    Rigid motions preserve angle measure between rays. All corresponding angle measures are preserved.

29. Which statement best matches SSS: $AB = DE$, $BC = EF$, $AC = DF$ or $AB = DE$, $\angle A = \angle D$, $AC = DF$?

    $AB = DE$, $BC = EF$, $AC = DF$. All three pairs of sides are given as equal.

30. What does it mean to "map" $A$ to $D$ under a rigid motion?

    The image of point $A$ is point $D$, written $A' = D$. The rigid motion sends point $A$ to point $D$.

31. In $\triangle ABC \cong \triangle DEF$, which side corresponds to $\overline{BC}$?

    $\overline{EF}$. Vertices correspond in the same order: $C \leftrightarrow F$.

32. In SAS, after mapping $\overline{AB}$ to $\overline{DE}$, what fixes the direction of $\overline{AC}$?

    The included angle equality $\angle A = \angle D$. The angle determines the ray direction from $A$.

33. Which criterion fails to guarantee congruence because it can change size: AAA or SSS?

    AAA. AAA creates similar triangles of different sizes.

34. Which angle is required in SAS: the included angle or a non-included angle?

    The included angle between the two sides. The angle must be between the two given sides.

35. What does "corresponding parts" mean after a rigid motion maps one triangle onto another?

    Images of vertices, sides, and angles match in the same order. Rigid motions preserve vertex-side-angle relationships.

36. What property of angle measure is preserved by any rigid motion?

    If $m\angle A = m\angle B$, then $m\angle A' = m\angle B'$. Angle measures stay the same under rigid motions.

37. What is the definition of triangle congruence using rigid motions?

    A rigid motion maps one triangle onto the other. One triangle is the image of the other.

38. Which congruence criterion uses two pairs of corresponding sides and the included angle?

    SAS. Uses two sides with their included angle.

39. Which congruence criterion uses two pairs of corresponding angles and the included side?

    ASA. Uses two angles with their included side.

40. What does SAS stand for in triangle congruence criteria?

    Side-Angle-Side. Two sides and the angle between them are equal.

41. What is a rigid motion (isometry) in the plane?

    A transformation that preserves all distances and angles. Maintains shape without stretching or shrinking.

42. What is the congruence criterion if $AB = DE$, $\angle B = \angle E$, and $BC = EF$?

    SAS (the included angle is $\angle B$). Angle $\angle B$ is between the two given sides.

43. What is the included angle between sides $AB$ and $AC$ in $\triangle ABC$?

    $\angle BAC$. The angle at vertex $A$ is between sides $AB$ and $AC$.

44. What is the congruence criterion if $AB = DE$, $BC = EF$, and $\angle A = \angle D$?

    None; this is SSA and does not guarantee congruence. The angle is not between the two given sides.

45. Identify the correct conclusion: If a rigid motion maps $\angle A$ to $\angle D$, then their measures satisfy what equation?

    $m\angle A = m\angle D$. Rigid motions preserve all angle measures.

46. In SSS, if $AB = DE$, $BC = EF$, and $AC = DF$, what is the conclusion?

    $\triangle ABC \cong \triangle DEF$ by SSS. All three corresponding sides are equal.

47. In an SSS setup, what geometric objects locate the third vertex from a fixed base $\overline{AB}$?

    Two circles centered at $A$ and $B$ with given radii. Circles centered at each endpoint with specified radii.

48. Identify the issue: $AB = DE$, $AC = DF$, and $\angle B = \angle E$ claimed as SAS.

    The angle is not included; this is SSA, not SAS. SAS requires the angle between the two given sides.

49. In ASA, after mapping $\overline{AB}$ to $\overline{DE}$, what determines the rays for the other sides?

    The equal angles at the endpoints: $\angle A$ and $\angle B$. Angles create rays that intersect at the third vertex.

50. What rigid motions can be used to map a segment $\overline{AB}$ onto $\overline{DE}$?

    A translation and rotation, possibly followed by a reflection. These transformations preserve all distances.

51. What is the minimum information needed to determine a triangle up to rigid motion: $3$ sides, $2$ sides and included angle, or $3$ angles?

    $3$ sides or $2$ sides with included angle; $3$ angles is insufficient. Three angles only determine shape, not size.

52. What is the included side between angles $\angle A$ and $\angle B$ in $\triangle ABC$?

    $\overline{AB}$. Side $AB$ connects vertices $A$ and $B$.

53. In SAS, if $AB = DE$, $AC = DF$, and $\angle A = \angle D$, what is the conclusion?

    $\triangle ABC \cong \triangle DEF$ by SAS. Two sides and included angle establish congruence.

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