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 complementary angle, supplementary angles and adjcent angles acute and obtuse angles types of angle measurement of an angle

Find the angles in each of the following.
The angle which is two times its complement

  1. $60^{\circ}$
  2. $120^{\circ}$
  3. $30^{\circ}$

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

Let the angle be $x^0$.


According to the question,
Complement angle $=(90^0-x)\times 2$

So,

$x=180^0-2x$

$3x=180^0$

$x=60^0$


Hence, this is the answer.

Multiple choice maths complementary angle, supplementary angles and adjcent angles acute and obtuse angles types of angle measurement of an angle

Find the angles in each of the following.
Two complementary angles are in the ratio $3 : 2$

  1. $54^{\circ}, 36^{\circ}$
  2. $44^{\circ}, 36^{\circ}$
  3. $54^{\circ}, 46^{\circ}$

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

Let the angles be $3x$ and $2x$.


So,
$3x+2x=90^0$

$5x=90^0$

$x=18^0$

Therefore, the angle are
$54^0, 36^0$


Hence, this is the answer.

Multiple choice maths complementary angle, supplementary angles and adjcent angles acute and obtuse angles types of angle measurement of an angle

If $\angle A$ is complement to $30^o$ and $\angle B $ is supplement to $120^o$ then:

  1. $\angle A > \angle B$
  2. $\angle A < \angle B$

  3. Incomparable
  4. $\angle A = \angle B$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Sum of complementary angles is $90^{\circ}$

$\Rightarrow \angle A+{ 30 }^{ \circ  }={ 90 }^{ \circ  }\ \Rightarrow \angle A={ 60 }^{ \circ  }$
Sum of supplementary angles is $180^{\circ}$
$\Rightarrow \angle B+{ 120 }^{ \circ  }={ 180 }^{ \circ  }\ \Rightarrow \angle B={ 60 }^{ \circ  }$
$\Rightarrow \angle A=\angle B$

Multiple choice maths complementary angle, supplementary angles and adjcent angles acute and obtuse angles types of angle measurement of an angle

Which is the greatest angle in the given set: $\dfrac{1}{3}$ of complete angle, $\dfrac{1}{3}$ of straight angle or a right angle?

  1. $\dfrac{1}{3}$ of complete angle
  2. $\dfrac{1}{3}$ of straight angle

  3. A right angle

  4. All are equal
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Complete angle $=360^{\circ}$

$\dfrac{1}{3}$ complete angle $=\dfrac { 1 }{ 3 } \times { 360 }^{ \circ  }={ 120 }^{ \circ  }$
Right angle $=90^{\circ}$
$\dfrac{1}{3}$ right angle $=\dfrac { 1 }{ 3 } \times { 90 }^{ \circ  }={ 30 }^{ \circ  }$
So $\dfrac{1}{3}$ complete angle is greater.

Multiple choice maths complementary angle, supplementary angles and adjcent angles acute and obtuse angles types of angle measurement of an angle

Rank the following angles in descending order. 
1. Straight angle
2. Reflex angle
3. Right angle 

  1. $2$, $1$, $3$
  2. $3$, $2$, $1$
  3. $2$, $3$, $1$
  4. $3$, $1$, $2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
  1. Right angle $=90^{\circ}$

    2. Straight angle $=180^{\circ}$

    3. Reflex angle lies between $180^{\circ}$ and $360^{\circ}$

    So the descending order of angles is
    $2>1>3$
Multiple choice maths complementary angle, supplementary angles and adjcent angles acute and obtuse angles types of angle measurement of an angle

In a triangle, the angles are in ratio $1: 3: 2$. Find the difference between the greatest and smallest angle of the triangle.

  1. $10^o$
  2. $70^o$
  3. $60^o$
  4. $20^o$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Le the angles be $x,3x$ and $2x$

Using angle sum property of triangle 
$x+3x+2x={ 180 }^{ \circ  }\ 6x={ 180 }^{ \circ  }\ \Rightarrow x={ 30 }^{ \circ  }$
So the angles are 
$x={ 30 }^{ \circ  }\ 3x=3\times { 30 }^{ \circ  }={ 90 }^{ \circ  }\ 2x=2\times { 30 }^{ \circ  }={ 60 }^{ \circ  }$
Difference between largest and smallest $={ 90 }^{ \circ  }-{ 30 }^{ \circ  }={ 60 }^{ \circ  }$

Multiple choice maths complementary angle, supplementary angles and adjcent angles acute and obtuse angles types of angle measurement of an angle

If the difference of two supplementary angles is $40^{\circ}$, then the measurement of the greater angle is

  1. $65^{\circ}$
  2. $110^{\circ}$
  3. $130^{\circ}$
  4. $220^{\circ}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the two supplementary angles are $x^{\circ}$ and
$180^{\circ} - x$
By hypothesis, $x - (180^{\circ} - x) = 40^{\circ}$
or $2x - 180^{\circ} = 40^{\circ}$
or  $2x = 220^{\circ}$
or    $x = 110^{\circ}$

Multiple choice maths complementary angle, supplementary angles and adjcent angles acute and obtuse angles types of angle measurement of an angle

In triangle $ABC,$ if $\dfrac { 1 }{ a+c } +\dfrac { 1 }{ b+c } =\dfrac { 3 }{ a+b+c } ,$ then $\angle c$  is equal to:

  1. $30^{\circ}$
  2. $45^{\circ}$
  3. $60^{\circ}$
  4. $90^{\circ}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given: $\cfrac { 1 }{ a+c } +\cfrac { 1 }{ b+c } =\cfrac { 3 }{ a+b+c } $


$\Rightarrow \quad \cfrac { a+b+2c }{ (a+c)(b+c) } =\cfrac { 3 }{ a+b+c } $

$ \therefore (a+b+2c)(a+b+c)=3(a+c)(b+c)$

$\Rightarrow$ $ { a }^{ 2 }+ab+ac+ab+{ b }^{ 2 }+bc+2ac+2bc+2{ c }^{ 2 }$$ =3(ab+ac+bc+{ c }^{ 2 })$

$ \therefore { a }^{ 2 }+2ab+3ac+{ b }^{ 2 }+3bc+2{ c }^{ 2 }$$ =3ab+3ac+3bc+3{ c }^{ 2 }$

$\Rightarrow$ ${ a }^{ 2 }+{ b }^{ 2 }=ab+{ c }^{ 2 }$

$\Rightarrow$ $ { a }^{ 2 }+{ b }^{ 2 }-{ c }^{ 2 }=ab$

$\Rightarrow$ $ \cfrac { { a }^{ 2 }+{ b }^{ 2 }-{ c }^{ 2 } }{ ab } =1$

$\Rightarrow \cfrac { { a }^{ 2 }+{ b }^{ 2 }-{ c }^{ 2 } }{ 2ab } =\cfrac { 1 }{ 2 } $

$\cos { C } =\dfrac{1}{2}\Rightarrow \angle C={ 60 }^{\circ}$

Multiple choice maths complementary angle, supplementary angles and adjcent angles acute and obtuse angles types of angle measurement of an angle

In $\Delta ABC$ and $\Delta DEF$, we have $\dfrac {AB}{DE}=\dfrac {BC}{FD}$. Triangles ABC and DEF will be similar if :

  1. $\angle A = \angle D$
  2. $\angle A = \angle F$
  3. $\angle B = \angle E$
  4. $\angle B = \angle D$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For two triangles to be similar by SAS similarity, the included angle between the proportional sides must be equal. Given AB/DE = BC/FD, the included angles are angle B and angle D, so angle B must equal angle D.