# Triangle Congruence Through Rigid Motions (Grade 10)

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## Triangle Congruence Through Rigid Motions

1.

If triangle ABC is congruent to triangle EDF, there must always exist some rigid transformations - reflections, rotations, and/or translations - that can map triangle ABC onto triangle EDF.

- True
- False

2.

You are given two triangles, ABC and FGH. There is a sequence of translations and reflections that map triangle ABC to triangle FGH, such that [math]bar{AB}[/math] maps to [math]bar{GH}[/math] and [math]bar{AC}[/math] maps to [math]bar{GF}[/math]. Which of the following statements is correct?

- There is not enough information to say if the triangles are congruent.
- [math]Delta ABC ~= Delta HGF[/math]
- [math]Delta ABC ~= FHG [/math]
- [math]Delta ABC ~= Delta GHF[/math]

3.

Triangle ABC undergoes the following transformations: a vertical translation of +5 units, a reflection about the x-axis, and a dilation by factor 2. The resulting triangle is labeled LMN. For the two triangles, we have [math]ang M ~= ang A[/math] and [math]angN ~= ang B[/math]. Which of the following is true?

- These triangles are not congruent.
- [math]Delta ABC ~= Delta MNL[/math]
- [math]Delta ABC ~= Delta NML[/math]
- [math]Delta ABC ~= Delta LNM[/math]

4.

If one triangle can be mapped onto another triangle through a sequence of rigid transformations, which of the following MUST be true? Choose all that apply.

- The name of the triangles is the same.
- The corresponding sides of the triangles are congruent.
- The corresponding angles of the triangles are congruent.
- The orientation of the triangles is the same.

5.

Triangle ABC, whose vertices are [math](-5,1), (-3,4),[/math] and [math](-2,1)[/math], is reflected over the y-axis and then rotated 90° clockwise about the origin. The points, in no particular order, resulting from the transformation are [math](1,-2), (1,-5),[/math] and [math](4,-3)[/math]. Labeling these points as F, G, and H respectively, which of the following is true?

- No relation between triangle ABC and transformed points.
- [math]Delta ABC ~= Delta FHG[/math]
- [math]Delta ABC ~= Delta HGF [/math]
- [math]Delta ABC ~= Delta GHF[/math]

6.

Which of the following transformations would show if [math]Delta ABC ~= Delta ADC ?[/math]

- Translate triangle ABC down by vertical distance between [math]bar{AC}[/math] and point B.
- Reflect triangle ABC over the line containing [math]bar{AC}[/math].
- Rotate triangle ABC 180° around midpoint of [math]bar{AC}[/math].
- Translate triangle ABC down by half the vertical length between [math]bar{AC}[/math] and point B, and then rotate it 180° about its centroid.

7.

Which of the following transformations would show if [math]Delta ABD ~= Delta CDB ?[/math]

- Translate triangle ABD right by the length of [math]bar{AD}[/math].
- Reflect triangle ABD over the line containing [math]bar{BD}[/math].
- Rotate triangle ABD 180° about the midpoint of [math]bar{BD}[/math].
- Reflect triangle ABD over the line containing [math]bar{BD}[/math] then translate [math]1/4 vec{BD}[/math].

8.

Which of the following transformations would show if [math]Delta ABF ~= Delta ECD ?[/math]

- Translate triangle ABF right by the length of [math]bar{FD}[/math].
- Reflect triangle ABF about the line joining the midpoints of [math]bar{BC}[/math] and [math]bar{FD}[/math].
- Rotate triangle ABF 180° about the midpoint of the line joining the midpoints of [math]bar{BC}[/math] and [math]bar{FD}[/math].
- Translate triangle ABF right by the length of [math]bar{FE}[/math].

9.

In the regular hexagon pictured, the center is M (not pictured). Which of the following transformations show that [math]Delta BMC ~= Delta DME ?[/math] Choose all correct answers.

- Reflect triangle BMC over the line joining the midpoints of [math]bar{CD}[/math] and [math]bar{AF}[/math].
- Rotate triangle BMC 120° counterclockwise about point M.
- Translate triangle BMC right the length of [math]bar{CD}[/math], then rotate it 120° clockwise about its centroid.
- Rotate triangle BMC 60° clockwise about point M, then reflect it over the line passing through C and F.

10.

In the diagram below, Shannon knows that [math]ang BAC ~= ang BCA ~= ang DFE ~= ang DEF[/math] and that [math]bar{BE}[/math] || [math]bar{AD}[/math]. To prove that the two triangles, ABC and FDE, are congruent, she proposes the following sequence of transformations to map [math]Delta ABC[/math] onto [math]Delta FDE[/math]: translate triangle ABC up by the length of the vertical distance between line segments [math]bar{BE}[/math] and [math]bar{AD}[/math], then translate triangle ABC right by the length of [math]bar{BF}[/math] plus half the length of [math]bar{FE}[/math], then reflect it over the line containing points B and E. Is this correct and why?

- No, the sequence of transformations is incorrect.
- No, you have to use a triangle congruence theorem.
- Yes, if the triangles map exactly onto each other (corresponding sides and angles are congruent), this proves [math]Delta ABC ~= Delta FDE[/math].
- It is unnecessary, since we can determine that [math]ang ABC ~= ang FDE[/math], and so already know the triangles are congruent.

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