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 triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

In $\Delta ABC, \angle B = 30^{\circ}, \angle C = 80^{\circ}$ and $\angle A = 70^{\circ}$ then,

  1. $AB > BC < AC$
  2. $AB < BC > AC$
  3. $AB > BC > AC$
  4. $AB < BC < AC$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In any triangle side opposite to the largest angle is the longest side.

Here $\angle C$ is largest and side opposite to it is $AB$
$\therefore AB$ is the longest side.
Then comes  $\angle A$ and side opposite to it is $BC$.
$\therefore BC$ is second longest.
Then comes $\angle B$ and side opposite to it is $AC$
So it is the smallest side.
So the decreasing order of sides is 
$AB>BC>AC$

Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

In $\Delta ABC$, if $\angle A = 50^{\circ}$ and $\angle B = 60^{\circ}$, then the greatest side is :

  1. AB

  2. BC

  3. AC

  4. Cannot say
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using angle sum property of triangle

$\angle A+\angle B+\angle C={ 180 }^{ \circ  }\ \Rightarrow { 50 }^{ \circ  }+{ 60 }^{ \circ  }+\angle C={ 180 }^{ \circ  }\ \Rightarrow \angle C={ 180 }^{ \circ  }-{ 110 }^{ \circ  }={ 70 }^{ \circ  }$
Side opposite to the largest angle is the longest side.
Here $\angle C$ is largest and side opposite to it is $AB$
So $AB$ is the longest side.

Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

In $\Delta ABC$, if $\angle A = 35^{\circ}$ and $\angle B = 65^{\circ}$, then the longest side of the triangle is :

  1. AC

  2. AB

  3. BC

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

Using angle sum property of triangle

$\angle A+\angle B+\angle C={ 180 }^{ \circ  }\ \Rightarrow { 35 }^{ \circ  }+{ 65 }^{ \circ  }+\angle C={ 180 }^{ \circ  }\ \Rightarrow \angle C={ 180 }^{ \circ  }-{ 100 }^{ \circ  }={ 80 }^{ \circ  }$
Side opposite to largest angle is the longest side of any triangle.
Here $\angle C$ is the largest angle and side opposite to it is $AB$
So $AB$ is the longest side.

Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

In $\Delta ABC$, if AB $>$ BC then :

  1. $\angle C < \angle A$
  2. $\angle C = \angle A$
  3. $\angle C > \angle A$
  4. $\angle A = \angle B$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In any triangle angle opposite to the longest side is largest.

Angle opposite to $AB$ is $\angle C$ and angle opposite to $BC$ is $A$
Here $AB>AC$
$\Rightarrow \angle C>\angle A$
So option $C$ is correct.

Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

In $\Delta ABC, \angle A=100^{\circ}, \angle B=30^{\circ}$ and $\angle C= 50^{\circ}$,then

  1. $AB>AC$
  2. $AB=AC$
  3. $AB<AC$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
In any triangle side opposite to the largest angle is the longest side.
Here $\angle A$ is largest and side opposite to it is $BC$
$\therefore BC$ is the longest side.
Then comes $\angle C$ and side opposite to it is $AB$
$\therefore AB$ is the second longest side.
Then comes $\angle B$ and side opposite to it is $AC$
$\therefore AC$ is the shortest side.
So the increasing order of sides is
$AC<AB<BC$
$\Rightarrow AB>AC$
So option $A$ is correct. 
Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

Which is the greatest side in the following triangle?
$\displaystyle \angle A:\angle B:\angle C=4:5:6$

  1. $AB$
  2. $BC$
  3. $AC$

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let $\angle A: \angle B: \angle C=4x:5x:6x$
$\therefore 4x+5x+6x=180$
$\therefore 15x=180$
$\therefore x=12$
Largest angle $=\angle C=6x=6\times 12=72$
Side opposite to greatest angle has greatest length. 
According to the given ratio, $\displaystyle \angle C$ is the greatest angle and thus$,$ $AB$ is the greatest side.

Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

The construction of a triangle $ABC$, given that $BC =$ $6$ cm, $B =$ $45 ^{\circ}$ is not possible when difference of $AB$ and $AC$ is equal to:

  1. $6.9$ cm
  2. $5.2$ cm
  3. $5.0$ cm
  4. $4.0$ cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

According to the theorem of inequalities, the sum of any two sides of the triangle is greater than the third side.

Therefore, $AC+BC>AB$
$\Rightarrow BC>AB-AC$
Therefore, only the first option that is $6.9$ cm does not satisfy the above equation. Rest all the options satisfy the equation.

Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

In triangle ABC, (b+c) cos A+(c+a)cos B+(a+b)cos C is equal to

  1. $0$
  2. $1$
  3. $a+b+c$
  4. $2(a+b+c)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$(b+c) \cos A+(c+a)\cos B+(a+b)\cos C$


$\Rightarrow$  $b\cos A+c\cos A+c\cos B+a\cos B+a\cos C+b\cos C$

$\Rightarrow$  $(b\cos C+c\cos B)+(c\cos A+a\cos C)+(a\cos B+b\cos A)$  ----( 1 )
Using projection formula,
$a=(b\cos C+c\cos B)$
$b=(c\cos A+a\cos C)$
$c=(a\cos B+b\cos A)$
Substituting above values in ( 1 ) we get,
$\Rightarrow$  $a+b+c$
$\therefore$   $(b+c) \cos A+(c+a)\cos B+(a+b)\cos C=a+b+c$

Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

Which statement is true about the difference of any two sides of a triangle?

  1. It is greater than the third side

  2. It is zero

  3. It is lesser than the third side

  4. It is lesser than zero
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let $a,b,c$ be the sides of triangle.

For constructing a triangle sum of any two sides must be greater than third side
$\Rightarrow a+b>c$
$\Rightarrow a>c-b$
$\Rightarrow c-b<a.......(i)$
Also $a+c>b$
$\Rightarrow  c>a-b$
$\Rightarrow a-b>c.....(ii)$
Also $b+c>a$
$\Rightarrow c>a-b$
$\Rightarrow a-b>c.......(iii)$
From $(i),(ii)$ and $(iii)$ it is clear that difference of any two sides is greater than the third side.
So option $C$ is correct.

Multiple choice maths acute and obtuse angles types of angle measurement of an angle angles in our surroundings

If one angle at a point is reflex angle, the other at that point may be :

  1. Acute angle

  2. Obtuse angle

  3. Straight angle

  4. Acute or obtuse angle
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Angle formed at a point is $\displaystyle { 360 }^{ o }$. 

When one of angle is reflex it range from $\displaystyle { 180 }^{ o }$ to $\displaystyle { 360 }^{ o }$. 
If angle is $\displaystyle { 200 }^{ o }$ then the other angle is $\displaystyle { 160 }^{ o }$, i.e obtuse angle. 
If one reflex angle is $\displaystyle { 300 }^{ o }$ then other angle is $\displaystyle { 60 }^{ o }$ i.e acute angle.

Multiple choice maths acute and obtuse angles types of angle measurement of an angle angles in our surroundings

Sum of two obtuse angle results in:

  1. Acute angle

  2. Right angle

  3. Obtuse angle

  4. Reflex angle
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Obtuse angle ranges from $\displaystyle { 90 }^{ o }$ to $\displaystyle { 180 }^{ o }$
So, sum of least obtuse angle is $\displaystyle { 91 }^{ o }+{ 91 }^{ o }={ 182 }^{ o }$ which is a reflex angle.
Sum of maximum obtuse angle is $\displaystyle { 179 }^{ o }+{ 179 }^{ o }={ 358 }^{ o }$ which is a reflex angle.

Acute angle is the angle which is greater than $\displaystyle { 0 }^{ o }$ and less than $\displaystyle { 90 }^{ o }$.

Obtuse angle is the angle which is greater than $\displaystyle { 90}^{ o }$ and less than $\displaystyle {180 }^{ o }$.
Right angle is the angle which is equal to $90^o$.
Reflex angle is the angle which is greater than $\displaystyle {180}^{ o }$ and less than $\displaystyle {360 }^{ o }$.

Multiple choice maths acute and obtuse angles types of angle measurement of an angle angles in our surroundings

Which of the following is a reflex angle?

  1. $\displaystyle { 180 }^{ o }$
  2. $\displaystyle { 360 }^{ o }$
  3. $\displaystyle { 204 }^{ o }$
  4. $\displaystyle { 135 }^{ o }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Reflex angle must be between $\displaystyle { 180 }^{ o }$ and $\displaystyle { 360 }^{ o }$. It can't be equal to $\displaystyle { 180 }^{ o }$ or $\displaystyle { 360 }^{ o }$ as these are straight and complete angle respectively.

Acute angle is the angle which is greater than $\displaystyle { 0 }^{ o }$ and less than $\displaystyle { 90 }^{ o }$.

Obtuse angle is the angle which is greater than $\displaystyle { 90}^{ o }$ and less than $\displaystyle {180 }^{ o }$.
Right angle is the angle which is equal to $90^o$.
Reflex angle is the angle which is greater than $\displaystyle {180}^{ o }$ and less than $\displaystyle {360 }^{ o }$.