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Quizzes > Mathematics > Geometry

Take the Similarity & Congruency Quiz Now!

Ready to ace tests of similarity? Try the geometry congruency test!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art geometric shapes and congruent figures on dark blue background for similarity congruency quiz

This Similarity & Congruency Quiz helps you practice tests of similarity, spot congruent figures, and solve triangle problems. Questions focus on ratios, scale factor, and when shapes match, so you can find gaps before a test. For extra practice, try the triangle congruence review or congruent triangles practice .

Which of the following is not a valid triangle congruency postulate?
SSS
ASA
AAA
SAS
The AAA (Angle-Angle-Angle) condition only establishes similarity, not congruency, because it does not fix the size of triangles. For congruency, a side must be used in at least one of the postulates (SSS, SAS, ASA, AAS or HL). AAA ensures angles are equal but allows triangles of different sizes. You can read more about triangle congruency postulates here: .
Two triangles having three pairs of equal angles are always:
Neither
Similar
Both congruent and similar
Congruent
When all three corresponding angles of two triangles are equal (AAA), the triangles are similar because their shape is identical but not necessarily their size. Similar triangles have proportional sides but may differ in scale. They are not necessarily congruent unless the corresponding sides happen to be equal in length. For more, see .
If two triangles are similar with a scale factor of 3, then the ratio of their areas is:
3 : 1
6 : 1
9 : 1
27 : 1
The ratio of areas of two similar figures is the square of the scale factor. Here, the scale factor is 3, so the area ratio is 3² = 9, giving 9 : 1. This works for any similar polygons. See more detail at .
In congruent triangles, corresponding sides and corresponding angles are:
Supplementary
Bisected
Equal
Proportional
Congruent triangles are identical in both shape and size, so each pair of corresponding sides and angles are exactly equal. Proportional sides and equal angles characterize similarity, not congruency. This distinction is fundamental in triangle geometry. For more, visit .
Which transformation always preserves size and shape of a figure?
Reflection
Enlargement
Dilation
Shear
A reflection is a rigid transformation that flips a figure across a line, preserving both size and shape. Dilation and enlargement change size, while a shear distorts shape. Thus only reflection (and also rotations and translations) guarantee congruency of the image. More at .
All squares are:
Isosceles
Similar
Congruent
Cyclic
All squares have four equal sides and four right angles, so any two squares are similar because their corresponding sides have the same ratio (1 : 1). They may differ in actual side length, so they are not necessarily congruent. Learn more on quadrilateral similarity at .
Which of the following is a valid similarity criterion for triangles (but not a congruency test)?
AAA
SAS
SSS
HL
AAA (Angle-Angle-Angle) ensures that all corresponding angles are equal, which is enough for similarity but not for congruency since side lengths can differ. SAS and SSS can serve as both congruency and similarity tests when used appropriately. HL is a right?triangle congruency test only. More info at .
A triangle is scaled by a factor of 1/2. If the original perimeter is 20 units, what is the new perimeter?
20 units
5 units
10 units
15 units
When you scale a figure by a factor k, its perimeter is multiplied by k. Here k = 1/2, so the new perimeter is 20 × 1/2 = 10. The same principle applies to any linear measure under dilation. More at .
Triangles with side lengths 3 , 4 , 5 and 6 , 8 , 10 are:
Congruent
Similar
Both congruent and similar
Neither
The second triangle is exactly twice each side of the first (6 = 2×3, 8 = 2×4, 10 = 2×5), so they are similar with scale factor 2. They are not congruent because their side lengths differ. More examples at .
In triangles ABC and DEF, if AB/DE = AC/DF and ?A = ?D, which similarity test applies?
SSS
ASA
AAS
SAS
When two sides are in proportion (AB/DE = AC/DF) and the included angles are equal (?A = ?D), the SAS (Side-Angle-Side) similarity postulate applies. This guarantees the triangles are similar. See for more.
Two squares have side lengths 4 and 6. What is the ratio of their areas?
4 : 6
16 : 36
6 : 4
2 : 3
Area of a square is side length squared. The smaller square has area 4² = 16 and the larger 6² = 36, so the ratio is 16 : 36, which simplifies to 4 : 9. Learn more at .
The scale factor between two similar rectangles is ?2. If the smaller has area 50, what is the larger area?
100
2
50?2
25
If linear scale factor is ?2, the area scale factor is (?2)² = 2. So the larger area is 50×2 = 100. This holds for any similar planar figures. More at .
If two congruent polygons have perimeters 20 and x, then x equals:
40
10
20
Cannot determine
Congruent polygons are identical in size and shape, so their perimeters must be equal. Therefore if one perimeter is 20, the other must also be 20. For more on congruent figures see .
In right triangle ABC with right angle at C, an altitude is drawn from C to hypotenuse AB, meeting it at D. Which is true about the smaller triangle ACD?
ACD is congruent to triangle ABC
ACD is similar to triangle BDC
ACD is similar to triangle ABC
ACD is congruent to triangle CBD
When you draw an altitude to the hypotenuse in a right triangle, the two smaller triangles are each similar to the original triangle by AA (they share angles). Triangle ACD shares ?A and the right angle at D, so it is similar to ABC. For more details see .
Given triangles ABC ? DEF with scale factor 2 : 1, if EF = 10, what is AB?
20
10
7.5
5
If triangle DEF is half the size of ABC (scale 2:1 from ABC to DEF), then EF corresponds to BC in the smaller. To find AB in the larger, you double EF: 10×2 = 20. However, note AB corresponds to DE, not EF, so we must match correctly. In fact AB/DE = 2, so if EF = 10 corresponds to BC, a similar calculation yields AB = 5 for DE. See .
In triangles ABC and DEF, the sides satisfy AB/DE = BC/EF = CA/FD = k, where k ? 1. What relationship exists between the triangles?
They are reflections of each other
They are congruent
They are not related
They are similar
If all three pairs of corresponding sides are in the same proportion k, then the triangles are similar by the SSS similarity criterion. They are not congruent unless k = 1. More at .
In a triangle, an angle bisector divides the opposite side into segments proportional to the adjacent sides. If AB = 8, AC = 6, and AD is the bisector of ?A meeting BC at D, what is BD/DC?
4 : 3
3 : 4
8 : 6
14 : 12
By the Angle Bisector Theorem, BD/DC = AB/AC = 8/6, which simplifies to 4/3. This is a key result in triangle geometry. See .
Two similar triangles have areas of 49 and 100. What is the ratio of their corresponding sides?
50 : 49
7 : 10
24 : 51
49 : 100
The ratio of areas equals the square of the ratio of corresponding sides. Taking the square root of 49/100 gives 7/10. Thus the side ratio is 7 : 10. Learn more at .
Given parallelogram ABCD, diagonal AC divides it into two congruent triangles. Which statement is true?
Area of ?ABC is half area of ?ADC
Area of ?ABC equals area of ?ADC
Area of ?ABC is twice area of ?ADC
Area of ?ABC equals area of the square on AB
In any parallelogram, a diagonal divides it into two congruent triangles, so they have equal area. This follows from SAS congruency (two sides and included angle). More at .
In triangle ABC, D, E, and F are midpoints of BC, AC, and AB respectively. What fraction of the area of ABC is the medial triangle DEF?
1/4
1/2
1/3
3/4
The medial triangle formed by connecting midpoints of the sides of any triangle has area equal to 1/4 of the original. Each side is half length, so area scales by (1/2)² = 1/4. More at .
In circle geometry, if two chords intersect inside a circle, the triangles formed by the chords are similar. This statement is:
Sometimes
True
False
Indeterminate
When two chords intersect inside a circle, they form vertical angles and inscribed angles that intercept the same arcs. By AA, the two triangles are similar. This is a standard result in circle theorems. See .
Given triangles ABC ? DEF, if AB = 10, DE = 5, EF = 15 and BC corresponds to EF, what is BC?
15
22.5
7.5
30
Scale factor from DEF to ABC is 10/5 = 2. Since BC corresponds to EF, BC = 2×15 = 30. Corresponding side ratios determine all lengths in similar triangles. More at .
In triangle ABC, point D is on BC such that BD/DC = AB/AC. What can be concluded about AD?
AD is an altitude
AD is an angle bisector
AD is a perpendicular bisector
AD is a median
The Angle Bisector Theorem states that the bisector of ?A divides the opposite side into segments proportional to the adjacent sides: BD/DC = AB/AC. Thus AD is the angle bisector. See .
Consider ?ABC with A(0,0), B(4,0), C(0,3) and ?DEF with D(0,0), E(8,0), F(0,6). These triangles are:
Congruent
Similar with scale factor 1/2
Similar with scale factor 2
Neither similar nor congruent
Triangle DEF has coordinates doubled from ABC (8=2×4, 6=2×3), so it's a dilation of factor 2 about the origin. The triangles are similar (not congruent) with scale factor 2. More at .
The midsegment in a triangle connects the midpoints of two sides. Which property is correct?
It's parallel to the third side and equal in length
It divides the triangle into two regions of equal area
It's parallel to the third side and half its length
It's perpendicular to the third side
The Triangle Midsegment Theorem states that the segment connecting midpoints of two sides is parallel to the third side and half as long. It also creates smaller similar triangles within. For details see .
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Study Outcomes

  1. Understand Tests of Similarity -

    Identify and recall the key criteria for determining similarity in geometric figures, such as AA, SSS, and SAS tests of similarity.

  2. Analyze Similar Triangle Conditions -

    Examine triangle pairs to decide if they meet similarity requirements using angle measures and side ratios.

  3. Apply Proportional Reasoning -

    Use ratios of corresponding sides to solve for unknown lengths in similar triangles quiz questions.

  4. Differentiate Congruency vs. Similarity -

    Distinguish when figures are congruent versus when they are merely similar by comparing side measurements and angles.

  5. Solve Geometry Congruency Test Problems -

    Employ congruency postulates like SSS, SAS, ASA, and HL to prove figures are congruent in geometry congruency test scenarios.

  6. Evaluate Real-World Applications -

    Apply learned concepts to practical scenarios by modeling situations with similar triangles and interpreting scale factors.

Cheat Sheet

  1. AA Criterion for Similarity -

    One of the most straightforward tests of similarity is the Angle-Angle (AA) postulate. If two pairs of corresponding angles in two triangles are equal, the triangles are similar, as outlined by Euclid and reinforced by Khan Academy's geometry materials. A handy mnemonic is "Two Angles Align, Similarity is Signed!"

  2. Side Ratios: SAS and SSS for Similarity -

    The SAS (Side-Angle-Side) and SSS (Side-Side-Side) tests of similarity use proportional ratios of corresponding sides. For SAS similarity, ensure two sides are in proportion and the included angle is congruent, while SSS similarity requires all three side pairs to maintain a consistent scale factor (ℓ : m = x : y = p : q). Practice these rules in a similar triangles quiz to internalize the scale factor concept.

  3. Proportional Reasoning and Scale Factors -

    Understanding that similar figures have side lengths in constant ratio is crucial for any similarity and congruency quiz. If ΔABC ~ ΔDEF with a scale factor of k, then AB = k·DE, BC = k·EF, and AC = k·DF, as detailed in the National Council of Teachers of Mathematics standards. A tip: write the ratio as AB/DE = BC/EF = AC/DF = k to keep your work organized.

  4. Congruency Tests: SSS, SAS, ASA, AAS -

    In a geometry congruency test, remember that congruent figures are identical in both shape and size. The four classic tests - SSS, SAS, ASA, and AAS - guarantee congruency when these conditions are met, per the University of Cambridge's math syllabus. Think "Some Snakes Always Slither Around Silently" to recall SSS, SAS, ASA, AAS!

  5. Congruency vs. Similarity: Key Differences -

    The main difference in a congruency vs similarity test is size preservation: congruent figures match exactly, while similar ones can be scaled. Congruent angles and equal sides define congruency, whereas similarity relies on angle equality and side proportionality, as explained by MIT OpenCourseWare. To remember, "Congruent = Copy, Similar = Shrink or Stretch."

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