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Quizzes > Mathematics > Geometry

Test Your Similar Triangles Skills: Unit 5 Practice Quiz

Dive into tests of similarity and master congruent/similar triangles now!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art layered triangles on coral background with text unit 5 practice test similar triangles answers

This Unit 5 practice quiz helps you find and check similar triangles answers, spot which triangles are similar, set up correct ratios, and solve for missing sides and angles. Get instant feedback and clear steps, then keep going with the Unit 5 geometry review or try more triangle congruence practice .

If two triangles are similar, what must be true about their corresponding angles?
They are complementary
They are parallel
They are equal in measure
They are supplementary
By definition, similar triangles have all corresponding angles equal in measure. This equality of angles ensures that the triangles maintain the same shape but possibly different sizes. Corresponding sides in similar triangles are proportional, but it is the angles that confirm the shape is the same. For more on angle relationships in similar triangles, see .
Which of the following is NOT a valid criterion for determining that two triangles are similar?
Side-Angle-Side (SAS)
Angle-Angle (AA)
Side-Side-Angle (SSA)
Side-Side-Side (SSS)
The AA, SAS, and SSS criteria are all valid tests for triangle similarity because they establish proportional sides or equal angles. However, SSA does not guarantee similarity or congruence because the two sides and a non?included angle do not fix the third side or the overall shape. SSA can lead to ambiguous cases in triangle construction. Learn more at .
Triangle ABC has side lengths 3, 4, and 5, and triangle DEF has side lengths 6, 8, and 10. What is the scale factor from triangle ABC to triangle DEF?
2
1.5
2.5
3
Each side of triangle DEF is exactly twice the corresponding side of triangle ABC (3?6, 4?8, 5?10), so the scale factor is 2. This proportional increase confirms the triangles are similar with a common ratio. Scale factors relate all corresponding linear measurements. Further explanation is available at .
If one triangle is similar to another with a scale factor of 3:1, what is the ratio of their areas?
6:1
1:3
3:1
9:1
The ratio of areas of two similar figures is the square of the scale factor. Here, (3:1)² = 9:1. That means the larger triangle's area is nine times the smaller's. This property holds for any pair of similar polygons. See for details.
When two triangles are similar, what is the relationship between the ratio of their perimeters and their scale factor?
Perimeter ratio equals scale factor
Perimeter ratio equals cube of scale factor
Perimeter ratio is the inverse of the scale factor
Perimeter ratio equals square of scale factor
For similar triangles, all linear measures, including perimeters, scale by the same factor. Thus the ratio of perimeters equals the scale factor itself. Area ratios use the square of the scale factor, but perimeters remain directly proportional. More on this can be found at .
In triangles ABC and DEF, if ABC ~ DEF, AB = 6, DE = 4, BC = 9, what is the length of EF?
9
6
4
8
Similarity gives AB/DE = BC/EF, so 6/4 = 9/EF. Solving EF = 9·4/6 = 6. This proportional approach finds corresponding side lengths. Learn more at .
A right triangle has legs of length 6 and 8. What is the length of the altitude from the right angle to the hypotenuse?
3.6
5.0
6.4
4.8
In a right triangle, the altitude to the hypotenuse equals the product of the legs divided by the hypotenuse: h = (6·8)/10 = 48/10 = 4.8. This follows from the similarity of the smaller triangles formed. See for the derivation.
A triangle with sides 5, 12, and x is similar to a triangle with sides 10, 24, and 26. What is x?
11
12
14
13
The scale factor from the smaller to larger triangle is 2 (5?10, 12?24), so the unknown side x must satisfy 2·x = 26, giving x = 13. Corresponding sides scale equally in similar triangles. More at .
If the ratio of corresponding sides of two similar triangles is 4:5, what is the ratio of their areas?
5:4
16:25
4:5
8:10
Area ratios are the square of linear ratios. Squaring 4:5 gives 16:25. Thus the larger triangle's area is 25 parts compared to 16 parts. This principle applies to all similar polygons. See .
A 5-foot tall person casts a 7-foot shadow at the same time a tree casts a 21-foot shadow. How tall is the tree?
10 feet
12 feet
18 feet
15 feet
Similar right triangles formed by the light rays give 5/7 = height_tree/21. Solving height_tree = 5·21/7 = 15 feet. Shadows and heights maintain proportionality in similar triangles. More examples at .
In ?ABC, DE is drawn parallel to BC with D on AB and E on AC. If AD = 4, DB = 6, and AC = 10, what is AE?
2
8
6
4
With DE ? BC, triangles ADE and ABC are similar, so AD/AB = AE/AC. Here AB = AD + DB = 10, giving 4/10 = AE/10, so AE = 4. Parallel segments create proportional lengths. See .
?ABC has sides AB = 8, AC = 6, BC = 10, and ?DEF ~ ?CBA. If DE = 5, what is the length EF?
7.5
3
6
4
The correspondence ?DEF ~ ?CBA maps DE to CB = 10, so scale factor = 5/10 = 1/2. EF corresponds to BA = 8, so EF = 8·(1/2) = 4. Correct correspondence of vertices is key. More at .
Triangle A(0,0), B(4,0), C(0,3) is similar to triangle A'(0,0), B'(2,0), C'(0,y). What is y?
2
1
2.5
1.5
The scale factor from triangle ABC to A'B'C' is 2/4 = 1/2. Applying that to the y?coordinate gives y = 3·(1/2) = 1.5. Corresponding coordinates scale by the same ratio. For coordinate similarity, see .
If ?ABC ~ ?DEF with ?A = ?D and ?B = ?E, AB = 10, DE = 5, AC = 14, what is DF?
7
8.4
4.2
6
Corresponding sides scale by AB/DE = AC/DF, so 10/5 = 14/DF gives 2 = 14/DF, hence DF = 7. Angle correspondence sets the correct side mapping. More on side ratios at .
In ?ABC, AD is the angle bisector of ?A meeting BC at D. If BD = 4, DC = 6, and AB = 10, what is AC?
15
20
12
24
By the Angle Bisector Theorem, AB/AC = BD/DC, so 10/AC = 4/6. Cross?multiplying yields AC = 10·6/4 = 15. This theorem directly relates adjacent side lengths on either side of an angle bisector. More detail at .
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Study Outcomes

  1. Identify Similar Triangle Postulates -

    Recognize and articulate the AA, SSS, and SAS criteria for similarity to accurately classify triangles in a similar triangles quiz.

  2. Apply Proportional Reasoning -

    Use ratios of corresponding sides to set up and solve equations, enabling you to determine missing lengths in tests of similarity.

  3. Differentiate Congruence and Similarity -

    Distinguish between congruent and similar triangles by analyzing side lengths and angle measures in congruent triangles practice scenarios.

  4. Compute Segment Lengths -

    Solve for unknown side lengths using properties of parallel lines and proportional segments within triangle diagrams.

  5. Analyze Complex Diagrams -

    Interpret overlapping and nested triangles to identify pairs of similar shapes and set up the corresponding similarity relationships.

  6. Evaluate Real-World Applications -

    Apply similarity concepts to solve practical geometry similarity test problems, from scale models to architectural layouts.

Cheat Sheet

  1. AA Similarity Postulate -

    The AA postulate states that two triangles are similar if they have two pairs of congruent angles. Recognizing matching angles quickly is essential for acing any geometry similarity test. Use the mnemonic "Angle-Angle Always Applies" to recall this criterion.

  2. SSS and SAS Similarity Criteria -

    The SSS and SAS criteria involve proportional sides and equal angles: if corresponding sides of two triangles are in proportion (SSS) or two sides are proportional and their included angle congruent (SAS), the triangles are similar. According to MIT OpenCourseWare, practicing these ratios on various tests of similarity will reinforce your ratio intuition. Remember the quick formula check: a/b = c/d for SSS and (a/b = c/d, ∠ included) for SAS.

  3. Triangle Proportionality Theorem -

    This theorem is a staple in any similar triangles quiz, stating that parallel lines cutting two sides of a triangle divide those sides proportionally. To find unknown lengths, set up ratios like AB/AC = DE/DF. Drawing auxiliary parallel lines as recommended by University of Cambridge resources helps visualize these proportional relationships.

  4. Scale Factor and Perimeter/Area Relationships -

    Once triangles are proven similar, the scale factor k relates corresponding sides (k = image/original) and dictates perimeters (P'=kP) and areas (A'=k²A). Applying these formulas in your unit 5 practice test-congruent/similar triangles answers will help you move from side lengths to area calculations seamlessly. Keep a quick-reference sheet of k and k² ratios for efficient computation on area similarity problems.

  5. Leveraging CPCTC Logic -

    While CPCTC formally applies to congruent triangles, understanding its reasoning aids in structuring similarity proofs by mapping corresponding parts. Use this approach in your congruent triangles practice to build airtight arguments, noting that angle and side pairings follow the same labeling logic once similarity is established. Always label triangles consistently (e.g., △ABC ∼ △DEF) to avoid vertex mix-ups during proofs.

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