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Quizzes > High School Quizzes > Mathematics

Figures Congruence & Similarity Practice Quiz

Sharpen skills with engaging geometric figure puzzles

Difficulty: Moderate
Grade: Grade 7
Study OutcomesCheat Sheet
Paper art representing a dynamic geometry quiz on congruence and similarity for high school students.

Easy
What does it mean for two figures to be congruent?
They have the same area only.
They are the same shape and size.
They have the same shape but different sizes.
They have equal angles but not necessarily equal side lengths.
Congruent figures have identical shapes and sizes, meaning every corresponding side and angle is equal. They can be mapped onto each other by rigid transformations such as rotations, reflections, or translations.
What defines two figures as similar?
They have the same shape but may differ in size.
They are rotated versions of each other with the same size.
They are identical in size and shape.
They have the same area but different shapes.
Similar figures have the same shape, which means their corresponding angles are equal and their sides are in proportion. However, they can vary in size depending on the scale factor.
Which transformation always produces a congruent image?
Shearing
Scaling
Rotation
Dilation
A rotation moves a figure about a fixed point without changing its size or shape, ensuring that the original and rotated images are congruent. Unlike dilations or scalings, a rotation does not alter the dimensions of the figure.
Which of the following best describes a reflection?
It creates a mirror image that is congruent to the original.
It rotates the figure by 180 degrees.
It creates an image with a reversed orientation that may differ in size.
It dilates the figure to double its size.
Reflection is an isometric transformation that produces a mirror image of the original figure while preserving both its size and shape. The only change is the reversal of orientation.
Which of the following best describes a dilation?
A transformation that changes only the size of a figure, preserving its shape.
A transformation that produces a mirror image.
A transformation that reflects a figure.
A transformation that rotates a figure.
A dilation enlarges or reduces a figure uniformly, changing its size while preserving its overall shape and the measures of its angles. The scale factor determines how much the figure is expanded or contracted.
Medium
If two similar triangles have corresponding side lengths in the ratio 3:4, what is the scale factor from the smaller triangle to the larger triangle?
4/3
1:1
7/3
3/4
The scale factor is the ratio of a side in the larger triangle to the corresponding side in the smaller triangle, which is 4/3. This means every side of the larger triangle is 4/3 times as long as the corresponding side of the smaller triangle.
Which property is always shared between any two congruent figures?
They have equal areas only.
They have the same area but different perimeters.
They have proportional side lengths.
They have equal corresponding angles and side lengths.
Congruent figures have exactly the same shape and size, which means both the corresponding sides and angles are identical. This is a key characteristic that distinguishes congruence from similarity.
Which aspect is not necessarily preserved during a dilation transformation?
Lengths of sides
Angles
Ratios of side lengths
Shape of the figure
During a dilation, the shape and the angle measures of the figure remain unchanged, and the ratios of corresponding side lengths are maintained. However, the actual lengths of the sides change based on the scale factor.
If two figures are similar but not congruent, what must be true?
They are identical in every way.
They have the same size but different shapes.
They have proportional areas but different angle measures.
They have the same shape but different sizes.
Similar figures share the same shape, meaning their corresponding angles are equal and their side lengths are proportional, even though their sizes may vary. This difference in size is the key factor that distinguishes similarity from congruence.
Which similarity criterion is most commonly used to prove two triangles are similar?
Right Angle-Hypotenuse-Angle (RHA)
Angle-Angle (AA)
Side-Angle-Side (SAS)
Side-Side-Side (SSS)
The AA criterion is widely used because if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. This method efficiently establishes the proportional relationships needed for similarity.
Which of the following sets of transformations are considered similarity transformations?
Reflections, rotations, and translations only
Translations, rotations, and shears
Translations, rotations, reflections, and dilations
Dilations only
Translations, rotations, and reflections are all isometries that preserve size and shape, and together with dilations, they are classified as similarity transformations because they maintain the proportionality of figures. These transformations ensure that corresponding angles remain equal.
If triangle ABC is congruent to triangle DEF, which of the following must be true?
Only the corresponding sides are equal.
Only the corresponding angles are equal.
Sides are proportional and angles are equal.
Both corresponding sides and angles are equal.
For two triangles to be congruent, every corresponding side and angle must match exactly. This is the defining property of congruence, ensuring that the triangles are identical in every geometric aspect.
In similar quadrilaterals, which constant ratio is shared between corresponding parts?
The ratio of the angles.
The ratio of corresponding side lengths.
The ratio of the diagonals.
The ratio of the areas.
Similar figures maintain a constant ratio between the lengths of corresponding sides. Although areas scale by the square of this ratio, the direct and primary proportionality lies in the side lengths.
How does a dilation with a scale factor of k affect the area of a figure?
The area is multiplied by k.
The area is multiplied by k squared.
The area remains unchanged.
The area is multiplied by 2k.
A dilation uniformly scales a figure's dimensions by a factor of k, which means its area, being a two-dimensional measure, is multiplied by k². This quadratic relationship is fundamental in understanding geometric scaling.
Which condition is not sufficient to prove that two triangles are congruent?
Side-Angle-Side (SAS)
Angle-Side-Angle (ASA)
Angle-Angle (AA)
Hypotenuse-Leg (HL) in right triangles
AA only establishes that the corresponding angles of two triangles are equal, proving similarity, not congruence. Without information about the side lengths, the triangles may differ in size despite having the same shape.
Hard
A triangle has side lengths in the ratio 2:3:4. A similar triangle has its shortest side of length 10. What are the side lengths of the similar triangle?
10, 12, 14
8, 12, 16
5, 7.5, 10
10, 15, 20
The original triangle's sides are in the ratio 2:3:4, so if the smallest side (2-part) is 10, the scale factor is 5. Multiplying each part by 5 gives side lengths of 10, 15, and 20, maintaining the original proportions.
Two similar parallelograms have corresponding side ratios of 5:8. What is the ratio of their areas?
64:25
5:8
8:5
25:64
For similar figures, the ratio of their areas is the square of the ratio of their corresponding sides. Here, squaring the side ratio (5/8) gives an area ratio of 25:64.
If a circle is dilated by a scale factor of 3, how are its circumference and area affected?
The circumference remains the same while the area is tripled.
The circumference is multiplied by 9 and the area is tripled.
The circumference is tripled and the area is multiplied by 9.
Both the circumference and area are tripled.
Dilation scales linear dimensions by the factor, so the circle's circumference is tripled. Since the area is two-dimensional, it is affected by the square of the scale factor, making it 9 times larger.
After applying a rotation followed by a dilation to a figure, which property is guaranteed to be preserved?
The orientation of the figure.
The lengths of sides.
The area of the figure.
The angle measures.
Rotation, being an isometry, preserves all angles and side lengths, while dilation preserves the angle measures despite changing side lengths. Thus, after both transformations, the angles remain unchanged.
If a triangle reflects over a line, an error occurs if one assumes the reflected triangle is only similar to the original. What is the mistake in this reasoning?
Overlooking that reflection is an isometry, which preserves size and shape.
Believing that reflection alters the angle measurements.
Confusing reflection with rotation.
Assuming that dilation has occurred during the reflection.
Reflection is an isometric transformation, meaning it preserves both the size and shape of the figure. The mistake is in assuming that similarity (which allows for size change) applies, when in fact the reflected image is congruent to the original.
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Study Outcomes

  1. Identify congruent figures by matching corresponding sides and angles.
  2. Differentiate between congruence and similarity by comparing shape and size.
  3. Analyze pairs of figures to determine if they exhibit congruent or similar properties.
  4. Apply geometric transformations to assess and establish figure congruence.
  5. Synthesize geometric information to justify claims about similarity between figures.

Congruent & Similar Figures Cheat Sheet

  1. Understanding Congruence - Congruent figures are like identical twins in geometry: same shape, same size, and every side and angle matches up perfectly. Visualize one shape turned, flipped, or slid, and it lands exactly on its twin. This concept is the foundation for proving more complex theorems. Learn more
  2. Recognizing Similarity - Similar figures share the same shape but can be different sizes, like photos taken with different zoom settings. Corresponding angles remain equal, while side lengths scale proportionally. It's a key tool for tackling real-world scaling problems with confidence. Learn more
  3. Triangle Congruence Criteria - Triangles can be proven congruent using Side-Side-Side (SSS), Side-Angle-Side (SAS), or Angle-Side-Angle (ASA) rules. Imagine matching three locks with three keys - once they all fit, you know you have the same triangle. These shortcuts save you from measuring every single angle and side! Learn more
  4. Triangle Similarity Criteria - To show triangles are similar, use Angle-Angle (AA), Side-Angle-Side (SAS), or Side-Side-Side (SSS) similarity tests. Think of resizing a favorite sports jersey; the shape stays the same, only the measurements change. Recognizing these shortcuts helps you find missing lengths faster. Learn more
  5. Proportionality in Similar Figures - In similar figures, all corresponding side ratios are equal, acting like a secret code for scaling problems. Use cross-multiplication to uncover unknown measurements with ease. It's a lifesaver when tackling tricky homework questions or exam puzzles! Learn more
  6. Angle Correspondence - Both congruent and similar shapes rely on matching angles to verify their relationships. Spotting equal angles is like finding matching puzzle pieces - it confirms you're on the right track. This principle keeps your geometric proofs solid and error-free. Learn more
  7. Scale Factor in Similarity - The scale factor is the magic multiplier that turns one similar figure into another. Multiply each side by this number to shrink or stretch shapes accurately. Mastering scale factors is crucial for resizing blueprints or model designs. Learn more
  8. Transformations and Congruence - Rigid motions - translations (slides), rotations (turns), and reflections (flips) - keep congruence intact by preserving shape and size. Picture moving a shape around on a whiteboard without stretching it. These moves are your toolkit for proving two figures are congruent. Learn more
  9. Transformations and Similarity - Similarity transformations combine rigid motions with dilations (resizing), changing size but keeping the same shape. It's like zooming in on a map while still recognizing the same streets and landmarks. Use this combo to demonstrate similarity topics in elegant proofs. Learn more
  10. Real-World Applications - Architects, engineers, and artists rely on congruence and similarity to create stable structures and stunning designs. From scaling down skyscraper blueprints to ensuring a sculpture's proportions pop, these concepts are everywhere. Understanding them gives you a ticket into countless STEM fields! Learn more
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