Triangles Test

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Triangles Test - 17

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Weekly Quiz Competition

Question 1
1 / -0

Two figures having the ............. shape but not necessarily the ............ size are called similar figures.

A
same, different

B
different, same

C
same, same

D
different, different

Solution

For two figures to be similar, their shape must be same but their sizes may be different.
For example, two circles of different radii, two equilateral triangles each having different lengths, etcetera.
Hence, two figures having the $$\underline {same}$$ shape but not necessarily the $$\underline {same}$$ size are called similar figures.
Question 2
1 / -0

For two triangles, if corresponding angles are equal, then the two triangles are similar. This is called ___ similarity.

A
AAA

B
SSS

C
SAS

D
none of these

Solution

For two triangles, If corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar. This is called $$AAA$$ similarity.
Option $$A$$ is correct.
Question 3
1 / -0

_____ congruent figures are similar but the converse is _______.

A
None, true

B
All, true

C
None, false

D
All, false

Solution

All congruent figures are similar but all similar figures are not congruent.
For eg. A pair of triangles which are similar by $$A.A.A.$$ test of similarity are not congruent pairs of triangles since the definite lengths of sides are unknown.
In $$\triangle ABC$$ and $$\triangle DEF$$, $$\angle A =\angle D= 50^o$$, $$\angle B =\angle E= 75^o$$ and $$\angle C =\angle F = 55^o$$.
Hence, $$\triangle ABC \sim \triangle DEF$$ but they are not congruent.
Question 4
1 / -0

For two triangles, if sides of one triangle are proportional to the sides of other triangle, then their corresponding angles are equal and hence the two triangles are similar. This is called ___ similarity.

A
$$AAA$$

B
$$SSS$$

C
$$SAS$$

D
none of these

Solution

For two triangles, if sides of one triangle are proportional to the sides of other triangle, then their corresponding angles are equal and hence the two triangles are similar. This is called $$SSS$$ similarity.
Option $$B$$ is correct.
Question 5
1 / -0

Complete the following sentence.
All squares are ___________.

A
Similar but not congruent

B
Congruent but not similar

C
Similar but may be congruent

D
Congruent but may be similar

Solution

Similarity means same shape, size may be different. All squares have same shape.
$$\therefore$$All squares $$are$$ similar.
Congruent means same shape and size. Different squares may have same or different sizes.
$$\therefore$$All squares $$may$$ be congruent.
Question 6
1 / -0

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

A
equal, same

B
unequal, same

C
equal, different

D
unequal, different

Solution

Let $$ABC$$ and $$DEF$$ be two triangles

Here, $$\angle A=\angle D$$

$$\angle B=\angle E$$

and $$\angle C=\angle F$$

That is, corresponding angles are equal

And their corresponding sides are in the same ratio i.e., $$\dfrac{AB}{DE}=\dfrac{BC}{EF}=\dfrac{CA}{FD}$$

Hence the two triangles are similar.
Question 7
1 / -0

In the figure, find $$\angle L$$.

A
$${60}^{o}$$

B
$${50}^{o}$$

C
$${70}^{o}$$

D
$${100}^{o}$$

Solution

From Given Figure,

$$\cfrac { AB }{ LM } =\cfrac { 4.4 }{ 11 } =\cfrac { 2 }{ 5 } $$

$$\cfrac { BC }{ MN } =\cfrac { 4 }{ 10 } =\cfrac { 2 }{ 5 } ;\quad \cfrac { CA }{ NL } =\cfrac { 3.6 }{ 9 } =\cfrac { 2 }{ 5 } $$

$$\Rightarrow \cfrac { AB }{ LM } =\cfrac { BC }{ MN } =\cfrac { CA }{ NL } $$

$$\Rightarrow \quad \triangle ABC\sim \triangle LMN$$ (SSS similarity)

$$\Rightarrow \angle L=\angle A\\={ 180 }^{ o }-\angle B-\angle C\\={ 180 }^{ o }-{ 50 }^{ o }-{ 70 }^{ o }\\={ 60 }^{ o } \ (\text{Using Angle Sum Property})$$

$$\therefore \angle L={ 60 }^{ o }$$
Question 8
1 / -0

If in the triangles $$ABC$$ and $$DEF$$, angle $$A$$ is equal to angle $$E$$, both are equal to $${40}^{o}$$, $$AB:ED=AC:EF$$ and angle $$F$$ is $${65}^{o}$$, then angle $$B$$ is:

A
$${35}^{o}$$

B
$${65}^{o}$$

C
$${75}^{o}$$

D
$${85}^{o}$$

Solution

In $$\triangle ABC$$ and $$\triangle DEF$$,
$$\angle A = \angle E = 40^{\circ}$$
$$\dfrac{AB}{ED} = \dfrac{AC}{EF}$$
Thus, $$\triangle ABC \sim \triangle DEF$$ (SAS rule)
or $$\angle C = \angle F = 65^{\circ}$$

Now, In $$\triangle ABC$$,
$$\angle A + \angle B + \angle C = 180$$ (Sum of angles of triangle)
$$40 + 65 + \angle B = 180$$
$$\angle B = 180 - 105$$
$$\angle B = 75^{\circ}$$
Question 9
1 / -0

G is the centroid of $$\bigtriangleup$$ABC. GE and GF are drawn parallel to AB and AC respectively. Find A($$\bigtriangleup$$ GEF) : A($$\bigtriangleup$$ABC).

A
$$1 : 9$$

B
$$1 : 3$$

C
$$1 : 16$$

D
$$1 : 4$$

Solution

Given: $$\triangle ABC$$, $$GE \parallel AB$$ and $$GF \parallel AC$$
Construction : Draw a median AD
Since, G is the centroid.
We know, $$\cfrac{GD}{AD} = \cfrac{1}{3}$$ (Centroid divided the median in the ratio of 2 : 1)
Ratio of areas of the triangle is equal to the ratio of the square of medians.
Thus, $$\cfrac{A(\triangle GEF)}{A(\triangle ABC)} = \cfrac{GD^2}{AD^2}$$
$$\cfrac{A(\triangle GEF)}{A(\triangle ABC)} = \cfrac{1}{9}$$
Question 10
1 / -0

In $$\triangle ABC$$ and $$\triangle DEF$$, $$\angle A={50}^{o}, \angle B={70}^{o}, \angle C={60}^{o}, \angle D={60}^{o}, \angle E={70}^{o}, \angle F={50}^{o}$$, then $$\triangle ABC$$ is similar to:

A
$$\triangle DEF$$

B
$$\triangle EDF$$

C
$$\triangle DFE$$

D
$$\triangle FED$$

Solution

In $$\triangle ABC$$ and $$\triangle DEF$$,
$$\angle A = \angle F = 50^{\circ} $$
$$\angle B = \angle E = 70^{\circ}$$
$$\angle C = \angle D = 60^{\circ}$$
thus, $$\triangle ABC \sim \triangle FED$$ (AAA rule)

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Triangles Test - 17

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Triangles Test - 17

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Question 1

same, different

different, same

same, same

different, different

Solution

Question 2

AAA

SSS

SAS

none of these

Solution

Question 3

None, true

All, true

None, false

All, false

Solution

Question 4

$$AAA$$

$$SSS$$

$$SAS$$

none of these

Solution

Question 5

Similar but not congruent

Congruent but not similar

Similar but may be congruent

Congruent but may be similar

Solution

Question 6

equal, same

unequal, same

equal, different

unequal, different

Solution

Question 7

$${60}^{o}$$

$${50}^{o}$$

$${70}^{o}$$

$${100}^{o}$$

Solution

Question 8

$${35}^{o}$$

$${65}^{o}$$

$${75}^{o}$$

$${85}^{o}$$

Solution

Question 9

$$1 : 9$$

$$1 : 3$$

$$1 : 16$$

$$1 : 4$$

Solution

Question 10

$$\triangle DEF$$

$$\triangle EDF$$

$$\triangle DFE$$

$$\triangle FED$$

Solution

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