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Quizzes > High School Quizzes > Mathematics

Unit 6 Similar Triangles Practice Quiz

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Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Colorful paper art promoting Similar Triangles Mastery trivia quiz for high school geometry students.

Which of the following criteria is sufficient to prove that two triangles are similar?
SSS (Side-Side-Side) Criterion
RHS (Right-angle Hypotenuse Side) Criterion
HL (Hypotenuse-Leg) Criterion
AA (Angle-Angle) Criterion
Triangles are similar if they have two pairs of congruent angles, satisfying the AA criterion. This is the simplest way to prove the similarity of triangles.
Which property is always true for similar triangles?
Their corresponding sides are equal in length.
They have congruent corresponding angles.
Their areas are always equal.
They have equal perimeters.
Similar triangles have all corresponding angles congruent, which is a defining property of similarity. Their sides, however, are only proportional, not equal in length.
In similar triangles, the scale factor is defined as:
The ratio of any two corresponding sides.
The product of the corresponding side lengths.
The difference between corresponding angles.
The sum of the corresponding side ratios.
The scale factor between similar triangles is obtained by dividing any side of one triangle by its corresponding side in the other triangle. This factor is key to relating the sizes of similar figures.
If triangle ABC is similar to triangle DEF, which of the following is a true statement?
AB - DE = BC - EF = AC - DF
AB/DE = BC/DF = AC/EF
AB + DE = BC + EF = AC + DF
AB/DE = BC/EF = AC/DF
In similar triangles, the ratios of corresponding sides are equal. This proportionality allows us to set up equations to solve for unknown lengths.
In similar triangles, the ratio of the perimeters is equal to:
The difference of the side lengths.
The cube of the scale factor.
The scale factor.
The square of the scale factor.
The perimeter is a linear measurement, so the ratio of the perimeters of similar triangles is equal to the scale factor between them. This is a direct consequence of the proportionality of corresponding sides.
Triangle ABC is similar to triangle DEF with a scale factor of 3. If side AB measures 4 units, what is the length of the corresponding side DE?
12 units
15 units
7 units
9 units
Multiplying the side length 4 by the scale factor 3 gives 12 units. This demonstrates how scale factors convert side lengths between similar triangles.
If the ratio of corresponding sides of two similar triangles is 2:5, what is the ratio of their areas?
2:25
4:25
5:2
2:5
The ratio of the areas of similar triangles is equal to the square of the ratio of their corresponding sides. Squaring 2:5 gives 4:25.
Two triangles have side lengths in the ratios 3:4:5 and 6:8:10 respectively. What can be concluded about these triangles?
They are isosceles.
They are congruent.
They have the same area.
They are similar.
The second triangle's sides are exactly twice those of the first, meaning all corresponding side ratios are equal. This confirms that the triangles are similar.
If two triangles are not similar, which of the following is a possible reason?
Their corresponding angles are not congruent.
They have the same shape.
Their corresponding angles are congruent.
Their corresponding sides are in proportion.
For triangles to be similar, all corresponding angles must be congruent. A failure of even one angle to match breaks the similarity condition.
In a pair of similar triangles, if an angle in one triangle measures 30°, what must the measure of the corresponding angle in the other triangle be?
30°
90°
60°
45°
Similar triangles have congruent corresponding angles. Therefore, the angle that measures 30° in one triangle must also measure 30° in the other.
Given triangle PQR is similar to triangle XYZ with a scale factor of 2, and side PQ measures 5 units, what is the length of the corresponding side XY?
2.5 units
7 units
5 units
10 units
Since the scale factor is 2, every side in triangle XYZ is twice as long as its corresponding side in triangle PQR. Multiplying 5 units by 2 yields 10 units.
Which property is a necessary condition to establish triangle similarity?
Two pairs of corresponding angles are congruent.
Two pairs of corresponding sides are equal.
Only one pair of corresponding angles are congruent.
The triangles have equal altitudes.
Two pairs of congruent corresponding angles are sufficient to establish triangle similarity via the AA criterion. This condition guarantees that the triangles have the same shape.
Two similar triangles have side lengths in the ratio 4:7. If the area of the smaller triangle is 16 square units, what is the area of the larger triangle?
49 square units
28 square units
64 square units
112 square units
The ratio of the areas of similar triangles is the square of the ratio of their sides. Squaring 4:7 gives 16:49, so the larger triangle's area is 49 square units.
In similar triangles, the medians correspond in what ratio?
They are inversely proportional.
They are always equal.
They follow the ratio of the areas.
They have the same ratio as the corresponding sides.
Medians, like all linear measurements in similar figures, scale with the same factor as the corresponding sides. This proportionality is a hallmark of similarity.
Which theorem is particularly useful in demonstrating altitude properties in right triangles using similar triangles?
Altitude-on-Hypotenuse Theorem
Angle Bisector Theorem
Law of Sines
Pythagorean Theorem
The Altitude-on-Hypotenuse Theorem uses the properties of similar triangles to relate the altitude of a right triangle to segments of the hypotenuse. This theorem is especially useful in geometric proofs involving right triangles.
In right triangle ABC with sides 3, 4, and 5, the altitude to the hypotenuse creates two smaller triangles similar to the original. If one segment of the hypotenuse is 2.25 units, what is the length of the corresponding segment in a similar triangle scaled by a factor of 2?
4.5 units
2.25 units
3.75 units
6.0 units
When a triangle is scaled by a factor of 2, every linear measurement, including segments on the hypotenuse, doubles. Therefore, a segment of 2.25 units becomes 4.5 units.
Two similar triangles have corresponding side lengths in the ratio 5:7. If the perimeter of the smaller triangle is 50 units, what is the perimeter of the larger triangle?
56 units
75 units
80 units
70 units
The perimeter scales linearly with the side lengths. Multiplying the smaller triangle's perimeter of 50 units by the scale factor of 7/5 gives a larger perimeter of 70 units.
In similar triangles XYZ and PQR, if side XY is 8 units, side XZ is 10 units, and the corresponding sides in triangle PQR are 12 and 15 units respectively, what is the length of the third side in triangle XYZ if the third side in triangle PQR is 18 units?
14 units
12 units
10 units
15 units
The scale factor from triangle XYZ to triangle PQR is consistent; 12/8 and 15/10 both equal 1.5. Therefore, the third side in triangle XYZ is 18 divided by 1.5, which equals 12 units.
Given two similar triangles with areas 36 and 81 square units respectively, what is the scale factor between their corresponding side lengths?
0.5
2
1.5
1.8
The scale factor is the square root of the ratio of the areas. Taking the square root of 81/36 gives 9/6, which simplifies to 1.5.
Triangle ABC has side lengths in the ratio 3:4:5. If a triangle similar to ABC has a perimeter of 72 units, what is the length of its shortest side?
18 units
21 units
20 units
24 units
The sum of the ratio parts is 3 + 4 + 5 = 12. Dividing the perimeter of 72 units by 12 gives 6, the scaling factor, and multiplying 6 by 3 yields 18 units for the shortest side.
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Study Outcomes

  1. Understand the criteria for triangle similarity, including AA, SAS, and SSS conditions.
  2. Identify and apply proportional relationships between corresponding sides of similar triangles.
  3. Analyze geometric proofs to verify the similarity of triangles.
  4. Solve for missing side lengths using properties of similar triangles.
  5. Apply similarity concepts to solve real-world and theoretical geometry problems.

Unit 6 Similar Triangles Test Answers PDF Cheat Sheet

  1. Understanding Similar Triangles - Similar triangles are like geometric twins: they share identical angle measures and their sides grow or shrink in perfect proportion, so they look the same but not necessarily the same size. Picture one triangle wearing a fun-sized T-shirt and the other rocking a giant jersey - they still match in shape! Math is Fun: Similar Triangles
  2. Angle-Angle (AA) Similarity Criterion - The AA rule is your ticket to triangle similarity when you only know angles: if two angles in one triangle match two angles in another, those triangles are buddies. No need for side lengths - just angle congruency seals the similarity deal! Math.net: AA Similarity
  3. Side-Angle-Side (SAS) Similarity Criterion - Imagine lining up two triangles so they share a matching angle and the sides around it are proportional; voila, similarity unlocked! This trick is awesome when you know two sides and the included angle but want to confirm they're in sync. Math.net: SAS Similarity
  4. Side-Side-Side (SSS) Similarity Criterion - When all three pairs of corresponding sides are proportional, you've struck gold with triangle similarity - no angles needed. Think of it as checking three puzzle pieces; if they all fit in the same ratio, the triangles are soulmates! Math.net: SSS Similarity
  5. Proportionality of Corresponding Sides - In similar triangles, sides are like best friends that grow at the same rate: if one is 3, 4, 5 the other could be 6, 8, 10 and they keep that exact 1:2 ratio! Spotting these proportional side lengths is your shortcut to proving similarity. Math Warehouse: Sides & Angles
  6. Angle Bisector Theorem - Slice a triangle's angle right down the middle and you'll see it chops the opposite side into segments that mirror the other two sides in proportion. It's like cutting a pizza so each slice has toppings in the same ratio - deliciously helpful! Math is Fun: Similar Theorems
  7. Side-Splitter Theorem - Draw a line parallel to one side of a triangle and watch it slice the other two sides into perfectly proportional pieces. This theorem is your backstage pass to solving for missing lengths when parallel lines are in play. Math is Fun: Similar Theorems
  8. Area Ratio of Similar Triangles - The area of similar triangles follows the square of the side ratio, so if sides scale by 2:1, areas zoom by 4:1 - think superhero strength! This fact supercharges your calculations when comparing triangle sizes. Math is Fun: Similar Theorems
  9. Applications of Similar Triangles - From figuring out the height of a towering tree by measuring its shadow to resizing maps for treasure hunts, similar triangles are your go-to tool. Mastering their magic makes real-world problems way less puzzling! Math is Fun: Applications
  10. Practice Problems - The best way to nail similarity is to dive into exercises: identify matching triangles, apply AA, SAS, SSS, and solve for those hidden sides or angles. Grab your pencil, flex those brain muscles, and watch your confidence soar! Math Planet: Triangle Practice
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