What are the four rules for similar triangles?

Two triangles are similar if they meet one of the following criteria. AA. : Two pairs of corresponding angles are equal. SSS. : Three pairs of corresponding sides are proportional. SAS. : Two pairs of corresponding sides are proportional and the corresponding angles between them are equal.

What is the symbol for similar triangles?

Difference between Similar Triangle and Congruent Triangle Similar Triangle Congruent Triangle They are symbolised by the sign ∼ They are symbolised by the sign ≅ All of the corresponding sides have the same ratio. The ratio of corresponding sides equal to a constant value 2 more rows May 3, 2023

Maths Quiz On Similar Triangles (2024)

1.
Which of the following triangles are always similar?
- A.
  Right triangle
- B.
  Obtuse triangle
- C.
  Equilateral triangle
- D.
  Isoceles triangle
Correct Answer
C. Equilateral triangle
Explanation
Equilateral triangles are always similar because all three sides are equal in length and all three angles are equal. This means that if you have two equilateral triangles, you can scale one up or down and rotate it, and it will still be congruent to the other triangle. Therefore, all equilateral triangles are similar to each other.
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30
2.
The sides of a triangle are 7,10, and 12. Find the length of the longest side of a similar triangle whose shortest side is 21.
- A.
  12
- B.
  21
- C.
  30
- D.
  36
Correct Answer
D. 36
Explanation
In a similar triangle, the corresponding sides are proportional. To find the length of the longest side of the similar triangle, we can set up a proportion using the given lengths of the sides.
Let x be the length of the longest side of the similar triangle.
Using the proportion:
7/21 = 12/x
Cross-multiplying:
7x = 21 * 12
Simplifying:
7x = 252
Dividing both sides by 7:
x = 36
Therefore, the length of the longest side of the similar triangle is 36.
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3.
In triangle ABC, angle A = 90º and angle B = 25º. In triangle DEF, angle E = 25º and angle F = 65º. Are the triangles similar?
- A.
  False
- B.
  True
Correct Answer
B. True
Explanation
Yes, the triangles are similar. Since angle B in triangle ABC is equal to angle E in triangle DEF, and angle A in triangle ABC is equal to angle F in triangle DEF, by the Angle-Angle (AA) similarity criterion, the triangles are similar.
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10
4.
Two triangles are similar. If the ratio of the perimeters is 5:3, find the ratio of the corresponding sides.
- A.
  5:3
- B.
  15:9
- C.
  25:9
- D.
  1:8
Correct Answer
A. 5:3
Explanation
If two triangles are similar, it means that their corresponding angles are equal, and the ratio of their corresponding sides is the same. In this case, the ratio of the perimeters of the triangles is given as 5:3. The perimeter of a triangle is the sum of its three sides. Since the ratio of the perimeters is 5:3, it means that the sum of the corresponding sides of the triangles is also in the ratio of 5:3. Therefore, the ratio of the corresponding sides of the triangles is 5:3.
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5.
Find the unknown length x in the following figure.
Correct Answer
15
6.
Which of the following conditionsare NOT sufficient to identify a pair of congruent triangles?
- A.
  Three pairs of corresponding sides are proportional (SSS).
- B.
  Two pairs of corresponding sides are proportional and their included angles are equal (SAS).
- C.
  Two pairs of corresponding sides are proportional and one pair of non-included angles are equal (SSA).
- D.
  Two pairs of angles are equal (AA).
Correct Answer
C. Two pairs of corresponding sides are proportional and one pair of non-included angles are equal (SSA).
Explanation
The condition that is NOT sufficient to identify a pair of congruent triangles is "Two pairs of corresponding sides are proportional and one pair of non-included angles are equal (SSA)." While the Side-Side-Angle (SSA) condition may suggest similarity, it alone is not enough to guarantee congruence between triangles.
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7.
What is true of angles of similar triangles?
- A.
  Opposite angles are congruent.
- B.
  Corresponding angles are congurent.
- C.
  Corresponding angles are 90 degrees.
- D.
  Corresponding angles are parallel
Correct Answer
B. Corresponding angles are congurent.
Explanation
In similar triangles, corresponding angles are congruent. This means that if two triangles are similar, the corresponding angles in each triangle have the same measure. This property is a key characteristic of similar triangles and is used to determine if two triangles are similar or not. Opposite angles being congruent, corresponding angles being 90 degrees, and corresponding angles being parallel are not true for all similar triangles.
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01
8.
Area of an equilateral triangle with side length a is equal to:
- A.
  √3/5a
- B.
  √3/4a
- C.
  √3/4 a²
- D.
  √2/3a
Correct Answer
C. √3/4 a²
Explanation
The correct answer, √3/4 a^2, represents the formula for finding the area of an equilateral triangle. In this formula, "a" represents the length of one side of the triangle. The square of "a" (a^2) is multiplied by the constant √3/4 to find the area. This formula is derived from the geometric properties of equilateral triangles and is a commonly used formula in mathematics.
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9.
If the perimeter of a triangle is 100 cm and the length of the two sides are 30 cm and 40 cm, the length of the third side will be:
- A.
  30
- B.
  45
- C.
  24
- D.
  16
Correct Answer
A. 30
Explanation
The perimeter of a triangle is the sum of the lengths of all its sides. In this case, the perimeter is given as 100 cm. The length of the two sides are given as 30 cm and 40 cm. To find the length of the third side, we subtract the sum of the lengths of the two given sides from the perimeter. So, 100 - (30 + 40) = 30 cm. Therefore, the length of the third side is 30 cm.
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10.
The height of an equilateral triangle of side 5 cm is:
- A.
  5 cm
- B.
  6.66 cm
- C.
  4.33 cm
- D.
  2.44 cm
Correct Answer
C. 4.33 cm
Explanation
The height of an equilateral triangle can be found using the formula h = (√3/2) * s, where h is the height and s is the length of a side. Plugging in the given side length of 5 cm into the formula, we get h = (√3/2) * 5 = 4.33 cm. Therefore, the correct answer is 4.33 cm.
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FAQs

AA (Angle-Angle): If triangles have two of the same angles, then the triangles are similar. SAS (Side-Angle-Side): If triangles have two pairs of proportional sides and equal included angles, then the triangles are similar.

If the two sides of a triangle are in the same proportion of the two sides of another triangle, and the angle inscribed by the two sides in both the triangle are equal, then two triangles are said to be similar. Thus, if ∠A = ∠X and AB/XY = AC/XZ then ΔABC ~ΔXYZ.

Angle Angle Angle (AAA)

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. It is sufficient to prove that only two pairs of angles are respectively equal to each other.

Two triangles are similar if they meet one of the following criteria.

AA. : Two pairs of corresponding angles are equal.
SSS. : Three pairs of corresponding sides are proportional.
SAS. : Two pairs of corresponding sides are proportional and the corresponding angles between them are equal.

The formula used to check if two triangles are similar or not depends on the condition of similarity. For two triangles △PQR and △XYZ , similarity can be proved using either of the following conditions, ∠P = ∠X, ∠Q = ∠Y and ∠R = ∠Z. PQ/XY = QR/YZ = PR/XZ.

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion . In other words, similar triangles are the same shape, but not necessarily the same size. The triangles are congruent if, in addition to this, their corresponding sides are of equal length.

In any 30-60-90 triangle, you see the following: The shortest leg is across from the 30-degree angle, the length of the hypotenuse is always double the length of the shortest leg, and you can find the length of the long leg by multiplying the short leg by the square root of 3.

Difference between Similar Triangle and Congruent Triangle

Similar Triangle	Congruent Triangle
They are symbolised by the sign ∼	They are symbolised by the sign ≅
All of the corresponding sides have the same ratio.	The ratio of corresponding sides equal to a constant value

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May 3, 2023

Similar triangle formulas are the formulas that tell us whether two triangles are similar or not. For two triangles △ABC and △XYZ, the similar triangles formula are, ∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z. AB/XY = BC/YZ = CA/ZX.

If the lengths of the corresponding legs of two right triangles are proportional, then by Side-Angle-Side Similarity the triangles are similar. In the figure, A B P Q = B C Q R . So, Δ A B C ∼ Δ P Q R .

Theorem 4.2.

Two triangles are similar if two angles of one equal two angles of the other (AA=AA). In Figure 4.2. 2, △ABC∼△DEF because ∠A=∠D and ∠B=∠E.

Maths Quiz On Similar Triangles (2024)

Which of the following triangles are always similar?

The sides of a triangle are 7,10, and 12. Find the length of the longest side of a similar triangle whose shortest side is 21.

In triangle ABC, angle A = 90º and angle B = 25º. In triangle DEF, angle E = 25º and angle F = 65º. Are the triangles similar?

Two triangles are similar. If the ratio of the perimeters is 5:3, find the ratio of the corresponding sides.

Find the unknown length x in the following figure.

Which of the following conditionsare NOT sufficient to identify a pair of congruent triangles?

What is true of angles of similar triangles?

Area of an equilateral triangle with side length a is equal to:

If the perimeter of a triangle is 100 cm and the length of the two sides are 30 cm and 40 cm, the length of the third side will be:

The height of an equilateral triangle of side 5 cm is:

FAQs

How do you prove triangles similar answers? ›

What is the formula for similar right triangles? ›

References