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 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.

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

Using ruler and compasses only, construct a triangle POR such that $\angle P = 120^{\circ}$, PO = 5 cm PR = 6 cm.In the same figure, find a point which is equidistant from its sides. Name this point With this point as centre draw a circle touching all the sides of the triangle.

  1. Circumcentre

  2. Incentre

  3. Mid point

  4. Data insufficient
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

(I) We draw a triangle POR with $\quad \angle RPO={ 120 }^{ O }.\quad $

(II) OI & OR are the angular bisectors of $\quad \angle RPO\quad &amp; \angle ROP\quad $
The bisectors intersect at I.
(i) IM & IN  are drawn perpendiculars from i to PR & PO respectively. 
 (iii) The circle which touches the sides of the triangle POR.
has the radius  IM=IN.
Justification-
Between $\quad \Delta IPM\quad &amp; \quad \Delta IPN,\ \angle IMP={ 90 }^{ o }=\angle INP,\ \angle IPM=\angle IPN\quad $
So the third angles  $\quad \angle PIN=\angle PIM\quad $
Also the side IP is common.
So, by ASA rule, $\quad \Delta IPM\equiv \Delta IPN.\quad $  
i.e IM=IN.
Similarly, by considering the triangles INO & ISO it can be shown that
IN=IS.
So IM=IN=IS.
i.e the circle with centre I touches the sides of the given triangle.
So  I is the INCENTRE of the triangle POR.
Ans- Option B.

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

The difference between two angles is $19$$\displaystyle ^{o}$ and their sum is $\displaystyle \frac{890}{9}^o$. Find the greater angle.

  1. $63^o$
  2. $35^o$
  3. $27^o$
  4. $59^o$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Let the two angles be $a$ and $b$

Then, 
$a  - b = 19$

and, $a + b = \dfrac{890}{9}$

Adding the two equations,

$2a = \dfrac{890 + 19\times 9}{9}$

$2a = \dfrac{1061}{9}$

Thus, $a = \dfrac{1061}{18}$

$a \approx 59^{\circ}$
Multiple choice maths complementary angle, supplementary angles and adjcent angles acute and obtuse angles types of angle measurement of an angle

If two angles of a triangle are acute angles, the third angle:

  1. is less than the sum of the two angles

  2. is an acute angle

  3. is the largest angle of the triangle

  4. may be an obtuse angle
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
For a triangle $ABC$, sum of angles is $180^0$
Two angles are given as acute, $\angle A, \angle B < 90^0$
$\angle A = 90^0 - x$
$\angle B = 90^0 - y$
where, $x,y < 90^0$
$\therefore \angle A + \angle B+ \angle C = 180^0$
$\Rightarrow \angle C = x+y $
$x+y$ can be $>90^0$ or $<90^0$
So, it may be obtuse or acute.
Hence, option D.
Multiple choice maths complementary angle, supplementary angles and adjcent angles acute and obtuse angles types of angle measurement of an angle

An angle which measures $\displaystyle 0^{o}$ is called:

  1. obtuse angle

  2. straight angle

  3. zero angle

  4. right angle
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

An angle which measure $0^0$ is called zero angle.

An angle which measure more than $0^o$ and less than $90^o$ is an acute angle.
An angle which measures $90^o$ is called a right angle.
An angle which measure more than $90^o$ and less than $180^o$ is an obtuse angle.

Hence, the answer is zero angle.