practice using the sas congruence theorem. which rigid transformations would map δjkl onto δpqr? select the three correct answers. □ a reflection only □ a rotation only □ a rotation and a reflection □ a translation and a reflection □ a translation only

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# Brief Explanations:
1. **Understanding Rigid Transformations**: Rigid transformations (reflection, rotation, translation) preserve the shape and size of a figure. We analyze how to map $ \triangle JKL $ to $ \triangle PQR $.
2. **Rotation and Reflection**: A rotation can align the triangles’ orientations, and a reflection (or vice versa) can flip/match them. A translation (shifting) combined with reflection or rotation can also work, but a single rotation or translation alone may not suffice due to orientation.
3. **Eliminating Incorrect Options**:
   - "A reflection only": Insufficient (needs alignment via rotation/translation).
   - "A rotation only": Insufficient (orientation may not match).
   - "A translation only": Insufficient (shifting alone won’t align angles/sides).
   - Correct options: A rotation and a reflection, a translation and a reflection, and (depending on orientation) a rotation (if angles align). Wait, re-evaluating: The triangles share a vertex at $ K $ (and $ Q $), so a rotation around $ K $ can align sides, then a reflection, or translation + reflection. The three correct are: *a rotation and a reflection*, *a translation and a reflection*, and (if rotation alone can align, but likely the three are rotation+reflection, translation+reflection, and maybe rotation? Wait, the diagram shows $ \triangle JKL $ and $ \triangle PQR $ with congruent sides (marked) and angles. So:
   - Rotation + Reflection: Rotate to align, reflect to flip.
   - Translation + Reflection: Translate, then reflect.
   - Rotation (if the angle of rotation aligns them, but maybe the three are: a rotation and a reflection, a translation and a reflection, and a rotation? Wait, no—standard for SAS congruence, the rigid motions to map one triangle to another:
   Correct options:
   - A rotation and a reflection
   - A translation and a reflection
   - (Wait, maybe "a rotation only" is not, but the three correct are: a rotation and a reflection, a translation and a reflection, and a rotation? No, recheck. The key is that the triangles are congruent, so rigid motions. Let’s list the options:
   - a reflection only: No, needs rotation/translation.
   - a rotation only: No, orientation may not match.
   - a rotation and a reflection: Yes.
   - a translation and a reflection: Yes.
   - a translation only: No.
   Wait, maybe the three are: a rotation and a reflection, a translation and a reflection, and a rotation? No, the correct three are: a rotation and a reflection, a translation and a reflection, and (maybe) a rotation? Wait, the problem says "select the three correct answers". Let’s re-express:
   - A rotation and a reflection: Yes (rotate to align, reflect to match).
   - A translation and a reflection: Yes (translate, then reflect).
   - A rotation (only): Wait, maybe if the rotation aligns them, but the diagram shows $ K $ and $ Q $ as common vertices, so rotating around $ K $ (to $ Q $) and then reflecting. Alternatively, the three correct are: a rotation and a reflection, a translation and a reflection, and a rotation? No, the standard answer for such problems is: a rotation and a reflection, a translation and a reflection, and (maybe) a rotation? Wait, no—let’s confirm:
   Rigid transformations: reflection, rotation, translation. To map $ \triangle JKL $ to $ \triangle PQR $:
   - Rotation + Reflection: Rotate to align sides, reflect to flip.
   - Translation + Reflection: Translate the triangle, then reflect.
   - Rotation (only): If the rotation aligns all sides/angles, but likely the three correct are: a rotation and a reflection, a translation and a reflection, and a rotation? No, the options are:
   1. a reflection only – No
   2. a rotation only – No
   3. a rotation and a reflection – Yes
   4. a translation and a reflection – Yes
   5. a translation only – No
   Wait, that’s two. Maybe I missed: the third is "a rotation only"? No, maybe the diagram has $ \triangle JKL $ and $ \triangle PQR $ with $ K $ and $ Q $ coinciding, so rotating around $ K $ (to $ Q $) and then reflecting, or translating and reflecting, or rotating. Wait, the correct three are: a rotation and a reflection, a translation and a reflection, and a rotation? No, the problem says "three correct answers". Let’s re-express:
   The triangles are congruent by SAS. To map $ \triangle JKL $ to $ \triangle PQR $:
   - Rotation: Rotate $ \triangle JKL $ around $ K $ to align with $ \triangle PQR $ (if angles match).
   - Rotation + Reflection: Rotate then reflect.
   - Translation + Reflection: Translate then reflect.
   So the three correct are:
   - a rotation and a reflection
   - a translation and a reflection
   - a rotation only (if rotation alone works)
   Wait, maybe the diagram shows $ \triangle JKL $ and $ \triangle PQR $ with $ KQ $ as a common side, so rotating around $ K $ (to $ Q $) and then reflecting, or translating and reflecting, or rotating. The correct options are:
   - a rotation and a reflection
   - a translation and a reflection
   - a rotation only (no, maybe not). Wait, the answer is:
   - a rotation and a reflection
   - a translation and a reflection
   - a rotation (only) – No, maybe the third is "a rotation only" is incorrect. Wait, the correct three are:
   3. a rotation and a reflection
   4. a translation and a reflection
   and another? Wait, maybe "a rotation only" is correct? No, let’s check the SAS congruence: the sides $ JK \cong PQ $, $ KL \cong QR $, and $ \angle JKL \cong \angle PQR $. So to map $ J \to P $, $ K \to Q $, $ L \to R $. So:
   - Rotation: Rotate $ \triangle JKL $ around $ K $ (to $ Q $) so that $ KL $ aligns with $ QR $, then $ JK $ aligns with $ PQ $.
   - Reflection: Reflect over a line to flip.
   - Translation: Translate $ \triangle JKL $ so that $ K $ maps to $ Q $, then reflect.
   So the three correct are:
   - a rotation and a reflection
   - a translation and a reflection
   - a rotation only (if rotation alone maps $ J \to P $, $ L \to R $)
   Wait, the options are:
   - a reflection only – No
   - a rotation only – Yes (if rotation around $ K $ maps $ J \to P $, $ L \to R $)
   - a rotation and a reflection – Yes
   - a translation and a reflection – Yes
   - a translation only – No
   So the three correct are: a rotation only, a rotation and a reflection, a translation and a reflection.

# Answer:
- a rotation only
- a rotation and a reflection
- a translation and a reflection

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