Mathematics · Quantitative Aptitude

Geometry of Triangles and Angles

758 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

When one acute angle of a triangle is equal to one acute angle of other triangle, and the triangles are right angles, do you think the triangles are similar?

  1. Not sure

  2. Similar

  3. Not similar

  4. Cannot be possible
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given one acute angle is equal and both triangles are right angled . Hence one angle of both triangles is $90^{\circ}$ each.

Hence the triangles are similar by $AA$ similarity criteria.
Option $B$ is correct.

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

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

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

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