Mathematics · Quantitative Aptitude

Geometry of Triangles and Angles

846 Questions

Triangle and angle geometry covers angle sums, properties of equilateral shapes, and right triangles. These principles form the basis of advanced quantitative aptitude sections. Practicing spatial problems helps secure points in exams.

Triangle angle sumsPythagorean theoremProperties of equilateral trianglesComplementary and supplementary anglesRight triangle formulas

Geometry of Triangles and Angles Questions

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

Which of the following is true?

  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.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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

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

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

  1. equilateral

  2. isosceles

  3. congruent

  4. not congruent
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

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

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

  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
Reveal answer Fill a bubble to check yourself
B Correct answer
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.

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

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

  1. equal, same

  2. unequal, same

  3. equal, different

  4. unequal, different
Reveal answer Fill a bubble to check yourself
A Correct answer
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. 

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

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.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

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

Which among the following is/are not correct ?

  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.
Reveal answer Fill a bubble to check yourself
A,B,C Correct answer
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 .

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

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.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $True$.

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

Two $\triangle sABC $ and DEF are similar. If $ar(DEF)= 243\ cm^2, ar(ABC)=108\ cm^2$ and $BC= 6\ cm$. Find $EF$.

  1. $9$
  2. $81$
  3. $3$
  4. $72$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given:-
$\triangle{ABC} \simeq \triangle{DEF}$
$ar \left( DEF \right) = 243 {cm}^{2}$
$ar \left( ABC \right) = 108 {cm}^{2}$
$BC = 6 cm$

To Find:- $EF = ?$

As we know that,
$\because \; \triangle{ABC} \simeq \triangle{DEF}$

$\cfrac{ar \left( \triangle{ABC} \right)}{ar \left( \triangle{DEF} \right)} = {\left( \cfrac{AB}{DE} \right)}^{2} = {\left( \cfrac{BC}{EF} \right)}^{2} = {\left( \cfrac{AC}{DF} \right)}^{2}$

$\therefore \; \cfrac{ar \left( \triangle{ABC} \right)}{ar \left( \triangle{DEF} \right)} = {\left( \cfrac{BC}{EF} \right)}^{2}$

$\Rightarrow \; \cfrac{108}{243} = \cfrac{{6}^{2}}{{EF}^{2}}$

$\Rightarrow \; {EF}^{2} = \cfrac{243}{108} \times 36$

$\Rightarrow \; EF = \sqrt{81}$

$\Rightarrow \; EF = 9$

Hence, the correct answer is $9$.
Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

STATEMENT - 1 : If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar.
STATEMENT - 2 : If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar.

  1. Statement - 1 is True, Statement - 2 is True, Statement - 2 is a correct explanation for Statement - 1

  2. Statement - 1 is True, Statement - 2 is True : Statement 2 is NOT a correct explanation for Statement - 1

  3. Statement - 1 is True, Statement - 2 is False

  4. Statement - 1 is False, Statement - 2 is True
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

 If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar.

If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar.
Both the statements are correct but $2$ is not the reason for $1$
If two corresponding angles are equal then the third corresponding become also equal , so the triangles are similar.
Option $B$ is correct

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

The ratio of the angles in $\triangle ABC$ is $2 : 3 : 4$. Which one of the following triangles is similar to $\triangle ABC ?$

  1. $ \triangle DEF $ has angles in the ratio $4 : 3 : 2.$
  2. $ \triangle PQR $ has angles in the ratio $1 : 2 : 3.$
  3. $ \triangle LMN $ has angles in the ratio $1 : 1 : 1.$
  4. $ \triangle STW $ has sides in the ratio $1 : 1 : 1.$
  5. $ \triangle XYZ $ has sides in the ratio $4 : 3 : 2.$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Similar triangles must have the same ratio of angles. The ratio 2:3:4 is equivalent to 4:6:8 or any scalar multiple, but the order of the ratio matters for similarity. Option A provides the same ratio 4:3:2, which represents the same set of interior angles as 2:3:4.

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

In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point in side AC such that AP = 6 cm. Prove that $\Delta\,BPC$ is similar to $\Delta\,ABC$. Also, find the length of BP.

  1. BP = 4 cm

  2. BP = 8 cm

  3. BP = 6 cm

  4. BP = 12 cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given: $\triangle ABC$, $AB = AC = 8$, $BC = 4$ and $AP = 6$

In $\Delta\,ABC$,
$\displaystyle\,\frac{AB}{BC}\,=\,\frac{8}{4}\,=\,2$,
In $\Delta\,BPC$,
$\displaystyle\,\frac{BC}{CP}\,=\,\frac{4}{2}\,=\,2$

Now, in $\triangle ABC$ and $\triangle BPC$
$\displaystyle\,\dfrac{AB}{BC}\,= \displaystyle\,\dfrac{BC}{CP}$
$\angle\,ABC\,=\,\angle\,C.$
Therefore, by SAS, $\Delta\,ABC \sim \Delta\,BPC$

Thus, $\dfrac{AB}{BP} = \dfrac{AC}{BC}$


$\dfrac{8}{BP} = \dfrac{8}{4}$
$BP = 4$ cm

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

In the given figure, $DE$ is parallel to $BC$ and the ratio of the areas of $\triangle ADE$ and trapezium $BDEC$ is $4:5.$ What is $DE : BC: ?$

  1. $1:2$
  2. $2:3$
  3. $4:5$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The ratio of area(ADE) to area(ABC) is 4/(4+5) = 4/9. Since the ratio of areas of similar triangles is the square of the ratio of their corresponding sides, DE/BC = sqrt(4/9) = 2/3.