Tag: maths

Questions Related to maths

If corresponding angles of two triangles are equal, then they are known as

maths congruence introduction to shapes similarity of triangles introduction to similar triangles

  1. Equiangular triangles

  2. Adjacent angles

  3. Supplementary angles

  4. Complementary angles

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Correct Option: A
Explanation:

If corresponding angles of two triangles are equal, then they are known as Equiangular triangles.

Option $A$ is correct.

If the angles of one triangle $ABC$ are congruent with the corresponding angles of triangle $DEF$, which of the following is/are true?

maths congruence introduction to shapes similarity of triangles introduction to similar triangles

  1. The two triangles are congruent but not necessarily similar.

  2. The two triangles are similar but not necessarily congruent.

  3. The two triangles are both similar and congruent.

  4. The two triangles are neither similar nor congruent.

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Correct Option: B
Explanation:

Only the angles of two triangles being congruent meaning the same, implies that the triangles are similar, since there could be many triangles having those angles but of varied sizes, just enlarging the sides in proportion.

For congruency, atleast one side has to be taken into account while writing the congruency test.

Which of the following is true?

maths congruence introduction to shapes similarity of triangles introduction to similar triangles

  1. The ratio of sides of two similar triangles is same as the ratio of their corresponding altitudes.

  2. The ratio of sides of two similar triangles is same as the ratio of their corresponding perimeters.

  3. The ratio of sides of two similar triangles is same as the ratio of their corresponding area

  4. The ratio of sides of two similar triangles is same as the ratio of their corresponding medians.

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Correct Option: A
Explanation:

Option A is correct as it is the property of similar triangle

If the area of two similar triangles are equal, then they are

maths congruence introduction to shapes similarity of triangles introduction to similar triangles

  1. equilateral

  2. isosceles

  3. congruent

  4. not congruent

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Correct Option: C
Explanation:

They are congruent.

$Consider\triangle ABC\quad and\triangle PQR$
$ \cfrac { ar(\triangle ABC) }{ ar(\triangle PQR) } =\cfrac { { AB }^{ 2 } }{ { PQ }^{ 2 } } =\cfrac { { AC }^{ 2 } }{ { PR }^{ 2 } } =\cfrac { { BC }^{ 2 } }{ { QR }^{ 2 } } $
$\implies\quad AB=PQ,\quad AC=PR,\quad BC=QR$
$ \therefore The\triangle ABC\quad$ and $\triangle PQR$  are congruent.

Two polygons of the same number of sides are similar if all the corresponding interior angles are:

maths congruence introduction to shapes similarity of triangles introduction to similar triangles

  1. Equal

  2. Proportional 

  3. Congruent

  4. Cannot say

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Correct Option: D
Explanation:

Two polygons of the same number of sides are similar, if: 

(a) Their corresponding angles are equal. 
(b) Their corresponding sides are in the same ratio (Proportional).
Hence, nothing can be said about two given polygons when only the angles are congruent, is known.

Triangle is equilateral with side$A$, perimeter $P$, area $K$ and circumradius $R$ (radius of the circumscribed circle). Triangle is equilateral with side $a$, perimeter $p$, area $k$, and circumradius $r$. If $A$ is different from $a$, then

maths congruence introduction to shapes similarity of triangles introduction to similar triangles

  1. $P : p = R : r$ only sometimes

  2. $P : p = R : r$ always

  3. $P : p = K : k$ only sometimes

  4. $R : r = K : k$ always

  5. $R : r = K : k$ only sometimes

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Correct Option: B
Explanation:

Since the triangles are similar, we have $A:a = P:p = R:r = \sqrt {K}: \sqrt {k}$ always, so that (b) is the correct choice.

If in two triangles, corresponding angles are _______ and their corresponding sides are in the ______ratio and hence the two triangles are similar.

maths congruence introduction to shapes similarity of triangles introduction to similar triangles

  1. equal, same

  2. unequal, same

  3. equal, different

  4. unequal, different

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Correct Option: A
Explanation:

If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar. 

State True or False
If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar.

maths congruence introduction to shapes similarity of triangles introduction to similar triangles

  1. True

  2. False

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Correct Option: A
Explanation:

.The statement is true if in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar by $AA$ similarity criteria.

Which among the following is/are not correct ?

maths congruence introduction to shapes similarity of triangles introduction to similar triangles

  1. The ratios of the areas of two similar triangles is equal to the ratio of their corresponding sides.

  2. The areas of two similar triangles are in the ratio of the corresponding altitudes.

  3. The ratio of area of two similar triangles are in the ratio of the corresponding medians.

  4. If the areas of two similar triangles are equal, then the triangles are congruent.

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Correct Option: A,B,C
Explanation:

The theorem is that the ratio of the areas of two similar triangles is equal to the square of the ratio of the corresponding sides.
In options A, B, and C this condition does not hold.
So option A, B, and C are not true.
But option D is true because if the areas of the similar triangles are equal then the sides will also be equal.
So, the triangles will be congruent by SSS test .

State True or False.
If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar.

maths congruence introduction to shapes similarity of triangles introduction to similar triangles

  1. True

  2. False

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Correct Option: A
Explanation:

If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar by $SAS$ similarity criteria.

Therefore the statement is $True$.