Middle School Math Quiz: Understand Parallel Line Transformation Properties
20 questions · exam conditions
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Understand Parallel Line Transformation PropertiesQuestion 1 of 20

A regular hexagon has three pairs of parallel sides. After a 60°60° clockwise rotation followed by a reflection across a line passing through two opposite vertices, which statement about the transformed hexagon is most accurate?

The hexagon still has exactly three pairs of parallel sides, but they are different pairs than in the original orientation.
The hexagon now has more than three pairs of parallel sides because the specific 60°60° rotation created additional parallelism.
The hexagon now has fewer than three pairs of parallel sides because the reflection broke some parallel relationships.
The hexagon still has exactly three pairs of parallel sides, and they are the same pairs as in the original hexagon.
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Middle School Math Quiz

Middle School Math Quiz: Understand Parallel Line Transformation Properties

Practice Understand Parallel Line Transformation Properties in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Understand Parallel Line Transformation Properties, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

A regular hexagon has three pairs of parallel sides. After a 60°60° clockwise rotation followed by a reflection across a line passing through two opposite vertices, which statement about the transformed hexagon is most accurate?

  1. The hexagon still has exactly three pairs of parallel sides, but they are different pairs than in the original orientation.
  2. The hexagon now has more than three pairs of parallel sides because the specific 60°60° rotation created additional parallelism.
  3. The hexagon now has fewer than three pairs of parallel sides because the reflection broke some parallel relationships.
  4. The hexagon still has exactly three pairs of parallel sides, and they are the same pairs as in the original hexagon. (correct answer)
Explanation: When you encounter transformation problems involving regular polygons, focus on a key principle: transformations like rotations and reflections preserve all geometric properties including parallel relationships. The specific measurements might change, but the fundamental structure remains identical. Let's work through this step-by-step. A regular hexagon has six equal sides arranged so that opposite sides are parallel, creating exactly three pairs of parallel sides. When you rotate this hexagon 60°60° clockwise, you're essentially moving each vertex to where the next vertex was positioned. Since 60°60° is exactly one-sixth of a full rotation (and the hexagon has 6-fold rotational symmetry), the hexagon maps perfectly onto itself. Following this with a reflection across a line through opposite vertices again preserves all parallel relationships—reflections maintain parallelism as a fundamental property. Choice A incorrectly suggests the parallel pairs change. While the hexagon's orientation changes, the same sides that were parallel initially remain parallel after both transformations. Choice B falls into the trap of thinking rotations can create new parallel relationships—they cannot. A regular hexagon always has exactly three pairs of parallel sides regardless of orientation. Choice C incorrectly assumes reflections can "break" parallelism, but reflections preserve all angle relationships and therefore all parallel relationships. The answer is D because both transformations preserve the hexagon's structure completely. The same three pairs of sides that were parallel before the transformations remain parallel afterward. Study tip: Remember that rigid transformations (rotations, reflections, translations) always preserve parallel relationships—they never create or destroy parallelism in geometric figures.

Question 2

A trapezoid has exactly one pair of parallel sides. This trapezoid undergoes a reflection across the xx-axis, then a translation, then a 270°270° counterclockwise rotation about point (0,0)(0, 0). After this sequence of transformations, how many pairs of parallel sides does the resulting figure have?

  1. Zero pairs, because the rotation changed the parallel relationship of the original sides.
  2. One pair, the same as the original trapezoid had before any transformations were applied. (correct answer)
  3. Two pairs, because the sequence of transformations converted the trapezoid into a parallelogram.
  4. The number depends on the specific measurements and orientation of the original trapezoid.
Explanation: Reflections, translations, and rotations are all rigid transformations that preserve parallel line relationships. Since the original trapezoid had exactly one pair of parallel sides, the transformed figure must also have exactly one pair of parallel sides. The definition of a trapezoid (exactly one pair of parallel sides) is preserved under any sequence of rigid transformations. Choice A incorrectly claims rotations destroy parallelism. Choice C incorrectly suggests transformations can change the fundamental shape properties. Choice D incorrectly implies that preservation of parallelism depends on specific measurements.

Question 3

Two horizontal lines are l1:y=3l_1: y=3 and l2:y=7l_2: y=7. They are reflected over the xx-axis to form l1′l_1' and l2′l_2'. Which statement is true?

  1. l1′l_1' and l2′l_2' are still parallel because both image lines are horizontal. (correct answer)
  2. l1′l_1' and l2′l_2' are not parallel because the slopes change to different values.
  3. l1′l_1' and l2′l_2' are perpendicular because reflection turns parallel lines into perpendicular lines.
  4. l1′l_1' and l2′l_2' intersect because one of them becomes vertical.
Explanation: This question tests understanding that rotations, reflections, and translations preserve parallel lines—if two lines are parallel before the transformation, their image lines remain parallel after. Parallel lines have equal slopes, indicating they point in the same direction: for example, y=2x+1 and y=2x+5 both have slope 2, so they are parallel and never intersect. Rigid transformations like reflections preserve slope relationships: reflecting y=3 and y=7 over the x-axis gives y=-3 and y=-7, both still with slope 0, so they remain parallel. For instance, points on l1 like (0,3) reflect to (0,-3) on y=-3, and similarly for l2, showing the image lines stay horizontal and parallel. The correct statement is that l1' and l2' are still parallel because both image lines are horizontal. A common error is claiming reflections change slopes to make lines perpendicular or intersecting, but all rigid transformations preserve parallelism. To verify, confirm the lines are initially parallel (both slope 0), apply the reflection to flip y-coordinates while keeping slopes zero, and check that image slopes are equal; this property holds because rigid transformations preserve angles, ensuring parallel relationships remain intact.

Question 4

Two lines on a coordinate plane are l1:y=2x+1l_1: y=2x+1 and l2:y=2x+5l_2: y=2x+5. A translation moves every point 3 units right and 2 units down (vector (3,−2)(3,-2)), creating image lines l1′l_1' and l2′l_2'. Are l1′l_1' and l2′l_2' parallel?

  1. No, because a translation changes the slope of each line.
  2. Yes, but only if the lines were horizontal or vertical to begin with.
  3. No, because the lines will intersect after moving right and down.
  4. Yes, because translations keep the slope the same, so the image lines still have equal slopes. (correct answer)
Explanation: This question tests understanding that rotations, reflections, and translations preserve parallel lines—if two lines are parallel before the transformation, their image lines remain parallel after. Parallel lines have equal slopes, indicating they point in the same direction: for example, y=2x+1 and y=2x+5 both have slope 2, so they are parallel and never intersect. Rigid transformations like translations preserve slope relationships: translating both lines by the vector (3,-2) results in new equations y=2x-1 and y=2x+3, where the slopes are still equal at 2, confirming they remain parallel. For instance, applying the translation to points on l1 and l2 shows the image lines l1' and l2' maintain the same direction and equal slopes before and after the shift. The correct answer is that l1' and l2' are parallel because translations keep the slopes the same and equal. A common error is thinking translations change slopes or cause lines to intersect, but all rigid transformations preserve parallelism, not just translations. To verify, confirm the lines are initially parallel (slopes both 2), apply the translation to adjust y-intercepts while keeping slopes unchanged, and check that the image slopes are still equal, proving parallelism is preserved; this property holds because rigid transformations maintain angles and directions, so parallel lines stay parallel.

Question 5

Lines ℓ1:y=3x+2\ell_1: y=3x+2 and ℓ2:y=3x−6\ell_2: y=3x-6 are parallel. Both lines are reflected across the yy-axis. What are the slopes of ℓ1′\ell_1' and ℓ2′\ell_2'?​

  1. Both slopes are 33.
  2. Both slopes are 13\tfrac{1}{3}.
  3. Both slopes are −3-3. (correct answer)
  4. The slopes are 33 and −3-3, so they are not parallel.
Explanation: This question tests understanding that rotations, reflections, and translations preserve parallel lines—if two lines are parallel before the transformation, their image lines remain parallel after. Parallel lines have equal slopes (same direction): y=3x+2 and y=3x-6 both have slope 3 so parallel (never intersect). Rigid transformations preserve slope relationships: reflecting both across the y-axis gives slopes -3 for both (still equal, still parallel). Reflection/rotation preserve parallelism similarly—parallel relationship (same angle with any transversal, equal slopes) maintains after any rigid transformation. For example, replacing x with -x in y=3x+2 gives y=-3x+2, and in y=3x-6 gives y=-3x-6, both with slope -3, remaining parallel. The correct answer is that both slopes are -3. A common error is thinking reflection inverts the slope differently for each line, but it affects them identically in terms of direction change.

Question 6

Lines ℓ1:x=−2\ell_1: x=-2 and ℓ2:x=5\ell_2: x=5 are parallel vertical lines. Both are rotated 180∘180^\circ about the origin. What are the equations of ℓ1′\ell_1' and ℓ2′\ell_2', and are they parallel?​

  1. ℓ1′:x=2\ell_1': x=2 and ℓ2′:x=−5\ell_2': x=-5; they intersect at the origin.
  2. ℓ1′:x=2\ell_1': x=2 and ℓ2′:x=−5\ell_2': x=-5; still parallel. (correct answer)
  3. ℓ1′:y=2\ell_1': y=2 and ℓ2′:y=−5\ell_2': y=-5; still parallel.
  4. ℓ1′:x=−2\ell_1': x=-2 and ℓ2′:x=5\ell_2': x=5; not parallel after rotation.
Explanation: This question tests understanding that rotations, reflections, and translations preserve parallel lines—if two lines are parallel before the transformation, their image lines remain parallel after. Parallel lines have equal slopes (same direction): vertical lines x=-2 and x=5 have undefined slope but are parallel (never intersect). Rigid transformations preserve slope relationships: rotating 180° about origin gives x=2 and x=-5 (still vertical, still parallel). Reflection/rotation preserve parallelism similarly—parallel relationship (same angle with any transversal, equal slopes) maintains after any rigid transformation. For example, points on x=-2 like (-2,0) rotate to (2,0), and on x=5 like (5,0) to (-5,0), resulting in vertical lines x=2 and x=-5 that remain parallel. The correct equations are x=2 and x=-5, and they are still parallel. A common error is assuming rotation makes vertical lines horizontal or intersecting, but 180° rotation inverts positions while keeping them vertical.

Question 7

A student says: "Parallel lines stay parallel after a translation, reflection, or rotation." Which reason best supports the student's statement?

  1. Only translations preserve angles, so reflections and rotations do not affect parallel lines.
  2. Rigid transformations always change slopes to different values, so lines cannot intersect.
  3. Rigid transformations preserve distances and angles, including the 0∘0^\circ angle between parallel lines. (correct answer)
  4. Parallel lines are defined as lines that are always the same distance apart, and that distance doubles after any transformation.
Explanation: This question tests understanding that rotations, reflections, and translations preserve parallel lines—if two lines are parallel before the transformation, their image lines remain parallel after. Parallel lines have equal slopes, indicating they point in the same direction: for example, y=2x+1 and y=2x+5 both have slope 2, so they are parallel and never intersect. Rigid transformations preserve these relationships by maintaining distances and angles, which define parallelism through equal corresponding angles with transversals. For instance, translating, reflecting, or rotating parallel lines keeps their directions consistent relative to each other. The best reason supporting the student's statement is that rigid transformations preserve distances and angles, including the 0° angle between parallel lines. A common error is claiming only translations preserve angles or that transformations change slopes, but all rigid ones preserve parallelism. To verify the property, note that parallelism relies on unchanged directional angles, which rigid transformations maintain universally; mistakes include confusing parallelism with constant distance, but it's about direction.

Question 8

Two parallel lines are l1:y=−x+2l_1: y=-x+2 and l2:y=−x−6l_2: y=-x-6. They are translated by (0,9)(0,9) (up 9 units) to form l1′l_1' and l2′l_2'. What are the slopes of l1′l_1' and l2′l_2'?

  1. The slopes are −1-1 and 99.
  2. The slopes are −1-1 and −9-9.
  3. Both slopes are 11.
  4. Both slopes are −1-1. (correct answer)
Explanation: This question tests understanding that rotations, reflections, and translations preserve parallel lines—if two lines are parallel before the transformation, their image lines remain parallel after. Parallel lines have equal slopes, indicating they point in the same direction: for example, y=2x+1 and y=2x+5 both have slope 2, so they are parallel and never intersect. Rigid transformations like translations preserve slope relationships: shifting up by 9 units keeps both slopes at -1 without change. For instance, the image equations are y=-x+11 and y=-x+3, both with slope -1, showing preserved parallelism. The correct answer is that both slopes are -1. A common error is miscalculating post-translation slopes as changed or unequal, but translations never alter slopes. To verify, confirm initial slopes are -1, apply the vertical shift which only affects y-intercepts, and check images have identical slopes; this preservation stems from rigid transformations maintaining line directions.

Question 9

Lines ℓ1:y=−12x+4\ell_1: y=-\tfrac{1}{2}x+4 and ℓ2:y=−12x−1\ell_2: y=-\tfrac{1}{2}x-1 are parallel. Both lines are rotated 90∘90^\circ counterclockwise about the origin. Which statement is true about ℓ1′\ell_1' and ℓ2′\ell_2'?

  1. They intersect because one line rotates into the other.
  2. They become perpendicular because any 90∘90^\circ rotation makes parallel lines perpendicular.
  3. They are still parallel because rotation preserves angles and direction relationships. (correct answer)
  4. They are no longer parallel because lines farther from the origin rotate by a different amount than lines closer to the origin.
Explanation: Rotations are rigid transformations, so they preserve angles and the relationships between lines. Since ℓ1\ell_1 and ℓ2\ell_2 both have slope −12-\tfrac12, they start out parallel, and rotating both by the same 90°90° turns both slopes into the same new value, so ℓ1′\ell_1' and ℓ2′\ell_2' stay parallel. Choice A is wrong because rotating both lines by the same angle keeps them distinct and parallel; it doesn't make one land on the other unless they already coincided. Choice B is wrong because rotating two parallel lines by the same angle keeps them parallel to each other, not perpendicular. Choice D is wrong because every point on a line rotates by the same angle around the center, regardless of how far it is from the origin, so distance from the origin doesn't change how much a line rotates.

Question 10

Three lines ℓ1:y=34x+1\ell_1: y=\tfrac{3}{4}x+1, ℓ2:y=34x−2\ell_2: y=\tfrac{3}{4}x-2, and ℓ3:y=34x+6\ell_3: y=\tfrac{3}{4}x+6 are parallel. All three are reflected across the xx-axis. Which describes the relationship among ℓ1′\ell_1', ℓ2′\ell_2', and ℓ3′\ell_3'?​

  1. The image lines become perpendicular to the original lines, so they cannot be parallel to each other.
  2. All three image lines are still parallel. (correct answer)
  3. The image lines are no longer parallel because reflection changes slopes by different amounts.
  4. Exactly two of the image lines are parallel; the third intersects them.
Explanation: This question tests understanding that rotations, reflections, and translations preserve parallel lines—if two lines are parallel before the transformation, their image lines remain parallel after. Parallel lines have equal slopes (same direction): all three have slope 3/4 so parallel (never intersect). Rigid transformations preserve slope relationships: reflecting across x-axis changes slopes to -3/4 for all (still equal, still parallel). Reflection/rotation preserve parallelism similarly—parallel relationship (same angle with any transversal, equal slopes) maintains after any rigid transformation. For example, y=(3/4)x+1 becomes y=-(3/4)x-1 after reflection, and similarly for others, all with slope -3/4, remaining parallel. The correct description is that all three image lines are still parallel. A common error is thinking reflection changes slopes by different amounts, but it uniformly negates them while preserving equality.

Question 11

A coordinate plane shows two parallel lines. Line ℓ1\ell_1 passes through (−4,−1)(-4,-1) and (0,1)(0,1), and line ℓ2\ell_2 passes through (−4,2)(-4,2) and (0,4)(0,4). Both lines are translated up 5 units. Which statement is true?

  1. The image lines intersect because moving up changes where they cross the yy-axis.
  2. The image lines become perpendicular because their slopes change after translation.
  3. The image lines are still parallel because translation does not change slope. (correct answer)
  4. Only one of the lines stays parallel because the translation affects lines differently.
Explanation: This question tests understanding that rotations, reflections, and translations preserve parallel lines—if two lines are parallel before the transformation, their image lines remain parallel after. Parallel lines have equal slopes (same direction): the given lines both have slope 1/2 so parallel (never intersect). Rigid transformations preserve slope relationships: translating both up 5 units keeps slopes 1/2 (still equal, still parallel). Reflection/rotation preserve parallelism similarly—parallel relationship (same angle with any transversal, equal slopes) maintains after any rigid transformation. For example, points (-4,-1) and (0,1) translate to (-4,4) and (0,6), with slope (6-4)/(0-(-4))=1/2, same for the other line, remaining parallel. The correct statement is that the image lines are still parallel because translation does not change slope. A common error is believing translation alters intersection or perpendicularity, but it only shifts without changing direction.

Question 12

Lines ℓ1\ell_1 and ℓ2\ell_2 are parallel, and line ℓ3\ell_3 intersects both of them. Transformation T1T_1 (a translation), then T2T_2 (a reflection), and finally T3T_3 (a rotation) are applied only to lines ℓ1\ell_1 and ℓ2\ell_2 together, while line ℓ3\ell_3 stays fixed. A student claims that lines ℓ1\ell_1 and ℓ2\ell_2 are no longer parallel to each other, because ℓ3\ell_3 now intersects them at different angles than before. Which analysis of this claim is correct?

  1. The student is correct because changing the intersection angles necessarily means the lines are no longer parallel to each other.
  2. The student is incorrect because transformations preserve all angle relationships, so intersection angles cannot change at all.
  3. The student is correct only if the rotation was not a multiple of 90°90°, since those rotations preserve intersection angles.
  4. The student is incorrect because rigid transformations preserve parallelism, even though intersection angles with transversals may appear different. (correct answer)
Explanation: Rigid transformations (translations, reflections, rotations) preserve all distance and angle relationships within the figure they are applied to. Since T1T_1, T2T_2, and T3T_3 are applied to ℓ1\ell_1 and ℓ2\ell_2 together, both lines move identically, so they stay parallel to each other. Line ℓ3\ell_3 is not transformed, so its angle with the new positions of ℓ1\ell_1 and ℓ2\ell_2 can look different than before, but that only reflects ℓ3\ell_3's relationship to the moved lines, not whether ℓ1\ell_1 and ℓ2\ell_2 are parallel to each other. Choice A confuses the transversal relationship with the parallel relationship. Choice B is incorrect because the transversal's angles can change since ℓ3\ell_3 itself was not transformed. Choice C is incorrect because parallelism between ℓ1\ell_1 and ℓ2\ell_2 is preserved regardless of the rotation angle used.

Question 13

In the figure, lines mm and nn are parallel. After applying a sequence of transformations consisting of a 90°90° clockwise rotation about point PP, followed by a reflection across line ℓ\ell, what can be concluded about the relationship between the images of lines mm and nn?

  1. The images are parallel because each individual transformation preserves parallelism, so their composition must also preserve parallelism. (correct answer)
  2. The images are perpendicular because the rotation changed their orientation by 90°90° relative to each other.
  3. The images may or may not be parallel depending on the specific location of point PP and line ℓ\ell.
  4. The images are parallel only if line ℓ\ell is perpendicular to both original lines mm and nn.
Explanation: Since rotations, reflections, and translations all preserve parallel lines, any composition of these transformations will also preserve parallelism. Lines mm and nn are parallel initially, so their images under any sequence of rigid transformations must remain parallel. Choice B incorrectly assumes the rotation affects the relationship between the lines rather than both lines equally. Choice C incorrectly suggests that the preservation of parallelism depends on the specific transformation parameters. Choice D incorrectly adds an unnecessary condition about line ℓ\ell.

Question 14

In the diagram, quadrilateral PQRSPQRS is shown with PQ‾∥SR‾\overline{PQ} \parallel \overline{SR} and PS‾∥QR‾\overline{PS} \parallel \overline{QR}. The quadrilateral is reflected across line kk, and then the image is rotated 45°45° about point TT. What is true about the sides of the final image quadrilateral?

  1. All sides will be parallel to the corresponding sides of the original quadrilateral PQRSPQRS.
  2. Exactly two pairs of opposite sides will be parallel to each other, maintaining the parallelogram property. (correct answer)
  3. Only one pair of opposite sides will remain parallel because the rotation disrupted one parallel relationship.
  4. No sides will be parallel to each other because the 45°45° rotation created non-parallel orientations throughout.
Explanation: The original quadrilateral PQRSPQRS is a parallelogram since both pairs of opposite sides are parallel. Rigid transformations (reflections and rotations) preserve all parallel relationships within a figure. Therefore, the final image will still be a parallelogram with exactly two pairs of parallel opposite sides. Choice A incorrectly describes the relationship between original and final positions rather than within the final figure. Choice C incorrectly suggests that rotations can destroy some but not all parallel relationships. Choice D incorrectly claims that rotations destroy parallelism entirely.

Question 15

A student says, "Parallel lines always stay parallel after a reflection, rotation, or translation." Which reason best supports the student's claim?​

  1. Because only translations preserve slope, and reflections and rotations do not.
  2. Because parallel lines are defined as lines that are exactly the same distance apart, and every transformation keeps distance the same for all figures.
  3. Because rigid transformations always change both slopes to 00.
  4. Because rigid transformations preserve angle measures, so lines that point in the same direction before a transformation still point in the same direction after. (correct answer)
Explanation: This question tests understanding that rotations, reflections, and translations preserve parallel lines—if two lines are parallel before the transformation, their image lines remain parallel after. Parallel lines have equal slopes (same direction): for example, y=2x+1 and y=2x+5 both slope 2 so parallel (never intersect). Rigid transformations preserve slope relationships: they maintain angles, including the directional alignment of parallels. Reflection/rotation preserve parallelism similarly—parallel relationship (same angle with any transversal, equal slopes) maintains after any rigid transformation. For example, with specific parallel lines, applying a reflection shows slopes either stay the same or change equally, keeping them parallel. The best reason is that rigid transformations preserve angle measures, including the 0° angle between parallel lines. A common error is confusing parallelism with constant distance, but parallelism is about direction, preserved by rigid motions.

Question 16

Three lines pp, qq, and rr are drawn such that p∥qp \parallel q and rr intersects both pp and qq. All three lines undergo a reflection across line ℓ\ell, then a 45°45° counterclockwise rotation about point OO, then a translation by vector v⃗\vec{v}. After these transformations, which relationship must be true?

  1. Lines p′p' and q′q' are no longer parallel because the 45°45° rotation changed their relationship.
  2. Line r′r' no longer intersects both p′p' and q′q', since the transformations changed how the lines relate.
  3. Lines p′p' and q′q' are parallel, and line r′r' intersects both p′p' and q′q'. (correct answer)
  4. The relationships depend on whether the 45°45° rotation was applied before or after the reflection.
Explanation: Reflections, rotations, and translations are all rigid transformations, so they preserve distances, angles, parallelism, and intersection relationships no matter how many are applied or in what order. Since p∥qp \parallel q before any of this, p′∥q′p' \parallel q' afterward, and since rr originally intersects both pp and qq, r′r' still intersects both p′p' and q′q'. Choice A is wrong because rotating by the same angle affects every line's slope the same way, so parallel lines stay parallel. Choice B is wrong because rigid transformations can't break an existing intersection. Choice D is wrong because the order of rigid transformations doesn't change which geometric relationships get preserved.

Question 17

Lines ℓ1:y=−12x+4\ell_1: y=-\tfrac{1}{2}x+4 and ℓ2:y=−12x−1\ell_2: y=-\tfrac{1}{2}x-1 are parallel. Both lines are rotated 90∘90^\circ counterclockwise about the origin. Which statement is true about ℓ1′\ell_1' and ℓ2′\ell_2'?

  1. They are no longer parallel because the 90∘90^\circ rotation changed the slope of each line by a different amount.
  2. They are still parallel because rotation preserves angles and direction relationships. (correct answer)
  3. They become perpendicular because any 90∘90^\circ rotation makes parallel lines perpendicular.
  4. They intersect because one line rotates into the other.
Explanation: Rotations are rigid transformations, so they preserve angles and the relationships between lines. Since ℓ1\ell_1 and ℓ2\ell_2 start with the same slope, −12-\tfrac{1}{2}, they're parallel, and rotating both by the same 90∘90^\circ turns both slopes into the same new value, so ℓ1′\ell_1' and ℓ2′\ell_2' stay parallel. Choice A is wrong because one rotation angle changes every line's slope the same way, not by different amounts for different lines. Choice C is wrong because rotating two parallel lines by the same angle keeps them parallel to each other; it doesn't make them perpendicular to each other. Choice D is wrong because rotating both lines together doesn't make one land on top of the other unless they already coincided before the rotation, which they don't here.

Question 18

In quadrilateral EFGHEFGH, side EF‾\overline{EF} is parallel to side HG‾\overline{HG}, but side EH‾\overline{EH} is not parallel to side FG‾\overline{FG}. After applying a rotation of 120°120° about point PP, followed by a reflection across line mm, what is true about the parallel relationships in the transformed quadrilateral?

  1. Exactly one pair of opposite sides remains parallel, and exactly one pair of opposite sides remains non-parallel. (correct answer)
  2. Both pairs of opposite sides are now parallel because the transformations created additional parallel relationships.
  3. No sides are parallel to each other because the 120°120° rotation disrupted the original parallel relationship.
  4. All sides are now parallel to their corresponding sides in the original quadrilateral position.
Explanation: The original quadrilateral has exactly one pair of parallel sides (EF‾∥HG‾\overline{EF} \parallel \overline{HG}) and one pair of non-parallel sides (EH‾\overline{EH} not parallel to FG‾\overline{FG}). Since rigid transformations preserve all geometric relationships, the transformed quadrilateral will maintain exactly the same parallel and non-parallel relationships as the original. Choice B incorrectly suggests transformations can create new parallel relationships. Choice C incorrectly claims rotations destroy existing parallelism. Choice D misunderstands the question by comparing positions rather than relationships within the transformed figure.

Question 19

Lines ℓ1\ell_1 and ℓ2\ell_2 are parallel. A student claims: "Only translations keep lines parallel; reflections and rotations can make parallel lines intersect." Which choice correctly evaluates the claim?

  1. The claim is false because parallel lines always become perpendicular after reflection
  2. The claim is true; reflections preserve parallelism but rotations do not
  3. The claim is false; translations, reflections, and rotations (rigid transformations) all preserve parallelism (correct answer)
  4. The claim is true; only translations preserve parallelism
Explanation: This question tests understanding that rotations, reflections, and translations preserve parallel lines—if two lines are parallel before the transformation, their image lines remain parallel after. Parallel lines remain so under all rigid transformations. The claim is false because reflections and rotations also preserve parallelism, not just translations. All rigid transformations maintain directions and angles. For example, reflecting or rotating parallel lines keeps them non-intersecting. Correct evaluation: claim false, all preserve; errors agree with claim. To verify: rigid transformations always preserve parallelism by maintaining equal slopes or angles.

Question 20

Lines ℓ1:y=3x+2\ell_1: y=3x+2 and ℓ2:y=3x−6\ell_2: y=3x-6 are parallel. Both lines are reflected across the yy-axis. What are the slopes of ℓ1′\ell_1' and ℓ2′\ell_2'?

  1. Both slopes are 33.
  2. The slopes are 33 and −3-3, so they are not parallel.
  3. Both slopes are −3-3. (correct answer)
  4. Both slopes are 13\tfrac{1}{3}.
Explanation: This question tests understanding that rotations, reflections, and translations preserve parallel lines—if two lines are parallel before the transformation, their image lines remain parallel after. Parallel lines have equal slopes (same direction): y=3x+2 and y=3x-6 both have slope 3 so parallel (never intersect). Rigid transformations preserve slope relationships: reflecting both across the y-axis gives slopes -3 for both (still equal, still parallel). Reflection/rotation preserve parallelism similarly—parallel relationship (same angle with any transversal, equal slopes) maintains after any rigid transformation. For example, replacing x with -x in y=3x+2 gives y=-3x+2, and in y=3x-6 gives y=-3x-6, both with slope -3, remaining parallel. The correct answer is that both slopes are -3. A common error is thinking reflection inverts the slope differently for each line, but it affects them identically in terms of direction change.