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Quizzes > High School Quizzes > Mathematics

Triangle Similarity Practice Quiz

Ace triangle applications with engaging practice problems

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Paper art representing a trivia quiz on triangle similarity for high school geometry students.

Which postulate is used to prove triangle similarity using two congruent angles?
Hypotenuse-Leg (HL)
Angle-Angle (AA)
Side-Side-Side (SSS)
Side-Angle-Side (SAS)
Two triangles are similar if they have two corresponding angles that are congruent. This is known as the AA postulate and is one of the most basic methods to establish similarity.
If two triangles are similar, which of the following ratios remains constant?
The sum of corresponding angles
The difference between corresponding side lengths
The ratio of corresponding side lengths
The product of corresponding side lengths
Similarity of triangles means that all corresponding sides have lengths in the same constant ratio. Therefore, the ratio of corresponding side lengths is preserved between similar triangles.
In similar triangles, which corresponding measure is congruent?
Corresponding medians
Corresponding sides
Corresponding circumferences
Corresponding angles
In similar triangles, all corresponding angles are congruent, although the sides are only proportional and not necessarily equal. This is a fundamental property of similar figures.
What does it mean for two triangles to be similar?
They have the same size and shape
Their corresponding angles are congruent and corresponding sides are proportional
They have equal perimeters
They have one pair of equal angles
Similarity implies that triangles have the same shape but not necessarily the same size. The defining property is that all corresponding angles are congruent and the sides are in proportion.
Triangle ABC ~ Triangle DEF with corresponding sides AB and DE, and AC and DF. If AB = 4, AC = 6, and DE = 2, what is the length of DF?
3
2
6
4
Since the triangles are similar, the ratios of corresponding sides are equal. Here, 4 divided by 2 equals 2, so DF must be 6 divided by 2, which is 3.
In triangles ABC and DEF which are similar, if side AB corresponds to DE with AB = 8 and DE = 4, and side BC corresponds to EF with BC = 10, what is the length of EF?
6
5
4
8
The similarity ratio from triangle ABC to DEF is 8/4 = 2, meaning each side in triangle ABC is twice the length of its counterpart in triangle DEF. Thus, EF is 10 divided by 2, which equals 5.
If two similar triangles have a ratio of corresponding side lengths of 3, what is the ratio of their areas?
9
12
3
6
The ratio of the areas of similar triangles is the square of the ratio of their corresponding side lengths. Since 3 squared is 9, the area ratio is 9.
Triangle PQR is similar to triangle STU. If the sides of PQR measure 3, 4, and 5 units and the shortest side of STU is 6 units, what is the scale factor from triangle PQR to triangle STU?
2
3
2.5
1.5
The scale factor is determined by dividing the corresponding side lengths. Since the shortest side of PQR is 3 and the corresponding side in STU is 6, the scale factor is 6/3 = 2.
In similar triangles, if the ratio of similarity is r, what is the ratio of their perimeters?
1/r
r
r²
It depends on the triangle's orientation
All corresponding linear measures in similar triangles, including perimeters, scale by the same factor r. Therefore, the ratio of the perimeters is simply r.
In triangle ABC ~ triangle DEF, if AB = 5 and DE = 10, what is the similarity ratio from triangle ABC to triangle DEF?
2
5
0.5
10
The similarity ratio is found by dividing a side of triangle DEF by the corresponding side of triangle ABC. In this case, 10 divided by 5 equals 2.
Which theorem explains that a line parallel to one side of a triangle divides the other two sides proportionally?
Base Angles Theorem
Pythagorean Theorem
Triangle Proportionality Theorem
Midsegment Theorem
The Triangle Proportionality Theorem states that if a line parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally. This theorem is fundamental in establishing the properties of similar triangles.
In similar triangles, if the lengths of sides in the smaller triangle are 2, 3, and 4 and the largest side in the larger triangle is 10, what is the scale factor?
2
3
2.5
4
The scale factor is determined by dividing a side of the larger triangle by its corresponding side in the smaller triangle. Here, the largest side scales from 4 to 10, so the scale factor is 10/4 = 2.5.
If triangles XYZ and LMN are similar and the ratio of their areas is 16:25, what is the ratio of their corresponding side lengths?
16:25
2:3
4:5
5:4
The ratio of corresponding side lengths in similar triangles is the square root of the ratio of their areas. Taking the square root of 16:25 yields 4:5.
Two similar triangles have corresponding side lengths in the ratio 7:9. If the perimeter of the smaller triangle is 42 units, what is the perimeter of the larger triangle?
48
56
54
63
Since the ratio of corresponding side lengths is 7:9, the perimeters also share this ratio. Multiplying 42 by 9/7 gives 54, which is the perimeter of the larger triangle.
Triangles ABC and DEF are similar. If AB = 9, AC = 12, DE = 3, and the side EF corresponding to AC is x, what is the value of x?
3
9
4
6
By the properties of similar triangles, the ratios of corresponding sides are equal. Since 9/3 equals 3, x must satisfy the equation 12/x = 3, which leads to x = 4.
In triangle ABC, line DE is drawn parallel to side BC, intersecting sides AB and AC at D and E respectively. If AD:DB = 2:3, what is the ratio of the area of triangle ADE to triangle ABC?
9/25
2/5
4/25
1/2
Since DE is parallel to BC, triangle ADE is similar to triangle ABC with a side ratio of AD/AB. Given AD:DB = 2:3, the full side AB is divided in a ratio of 2 to 5, so the ratio is 2/5. Squaring this gives (2/5)² = 4/25 for the area ratio.
A ladder leaning against a wall forms a right triangle. A line drawn parallel to the ground creates a smaller similar triangle within the larger one. If the ratio of the vertical sides of the small to the large triangle is 3:5, and the full vertical height of the large triangle is 15 ft, what is the height of the smaller triangle?
9 ft
6 ft
12 ft
10 ft
The ratio of the vertical sides of the small to the large triangle is 3:5. Therefore, the height of the smaller triangle is (3/5) multiplied by 15 ft, which equals 9 ft.
In two similar triangles, if a side in the smaller triangle is 7 units and the corresponding side in the larger triangle is 21 units, and the area of the smaller triangle is 35 square units, what is the area of the larger triangle?
105
210
420
315
The scale factor between the triangles is 21/7 = 3. Since area scales by the square of the scale factor, the larger triangle's area is 35 multiplied by 3², which is 35 x 9 = 315 square units.
A rectangle is inscribed in a right triangle such that one side lies on the triangle's base. A line through the rectangle's opposite vertex is drawn parallel to the hypotenuse, creating a smaller triangle. Which concept does this configuration demonstrate?
Triangle Proportionality
The Pythagorean Theorem
The Angle Bisector Theorem
Properties of Similar Medians
Drawing a line parallel to the hypotenuse creates smaller triangles within the larger triangle that are similar to it. This configuration demonstrates the principle of triangle proportionality, where corresponding segments are divided in proportion.
In triangle ABC, points D and E on sides AB and AC, respectively, are such that DE is parallel to BC. If the area of triangle ABC is 72 square units and that of triangle ADE is 18 square units, what is the ratio AD:AB?
1:6
1:2
1:4
1:3
Since triangle ADE is similar to triangle ABC, the ratio of their areas equals the square of the ratio of corresponding sides. With an area ratio of 18/72 = 1/4, taking the square root gives a side ratio of 1/2, which means AD:AB is 1:2.
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Study Outcomes

  1. Analyze the criteria for establishing triangle similarity.
  2. Identify similar triangles in a variety of geometric configurations.
  3. Apply proportional reasoning to solve for unknown side lengths in similar triangles.
  4. Evaluate relationships between angles in similar triangles to determine congruency.
  5. Utilize triangle similarity concepts to solve real-world geometry problems.

Triangle Similarity Quiz: Applications Cheat Sheet

  1. Similar Triangles Basics - Similar triangles have equal corresponding angles and proportional sides, meaning they're the same shape in different sizes. Get ready to spot patterns like a geometry detective! Dive into Similar Triangles 101
    2. GeeksforGeeks
  2. Angle‑Angle (AA) Criterion - If two angles of one triangle match two angles of another, those triangles are similar - no side measurements needed! It's the quickest trick to confirm similarity and save yourself some calculations. Master AA Similarity
    3. BYJU'S
  3. Side‑Angle‑Side (SAS) Criterion - When two sides of one triangle are proportional to two sides of another and their included angles are equal, voilà - they're similar! Perfect for when you only know two sides and the angle between them. Crack SAS Similarity
    4. BYJU'S
  4. Side‑Side‑Side (SSS) Criterion - If all three sides of two triangles are in proportion, those triangles are similar - full stop. Use this when you've got every side length at your fingertips. Explore SSS Similarity
    5. BYJU'S
  5. Area Ratio Rule - The ratio of areas of similar triangles equals the square of their corresponding side ratio, so a 2:1 side ratio gives a 4:1 area ratio! This neat square trick makes area problems a breeze. Learn Area Ratios
    6. BYJU'S
  6. Basic Proportionality Theorem - Draw a line parallel to one side of a triangle, and it fragments the other sides proportionally. It's a cornerstone for proving other similarity cases. See Thales' Theorem in Action
    7. GeeksforGeeks
  7. Angle Bisector Theorem - An angle bisector splits the opposite side into segments proportional to the adjacent sides. This handy tool can simplify many tricky triangle problems. Angle Bisector Breakdown
    8. MathsIsFun
  8. Equilateral Triangle Special Case - All equilateral triangles are inherently similar because all angles and sides match perfectly in proportion. It's the easiest similarity scenario out there! Equilateral Similarity Explained
    9. BYJU'S
  9. Congruent vs. Similar - Congruent triangles are similar triangles with identical side lengths, not just proportional ones. Remember: every congruent triangle is similar, but not every similar triangle is congruent! Spot the Difference
    10. GeeksforGeeks
  10. Practice Problems - Tackling a variety of problems cements your understanding and boosts confidence. Grab your pencil and challenge yourself with real examples! Try Practice Questions
    11. GeeksforGeeks
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