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 when lines join trapeziums and kites quadrilaterals and their properties closed figures

Given a trapezium ABCD in which $AB||CD$ and $AD=BC$. If $\angle C=76^{\circ}$, then $\angle D$ equals

  1. $14^{\circ}$
  2. $104^{\circ}$
  3. $76^{\circ}$

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

In trapezium ABCD, $AB \parallel CD$, $AD = BC$
Draw a perpendicular from A on CD to meet CD at M and a perendicular from B on CD to meet at N.
Now, in $\triangle ADM$ and $\triangle BNC$
$\angle AMD = \angle BNC$ (Each $90^o$)
$AM = BN$ (distance between parallel lines)
$AD = BC$ (Given)
Thus, $\triangle ADM \cong \triangle BCN$ (SAS rule)
Hence, $\angle D = \angle C = 76^{\circ}$ (by CPCT)

Multiple choice maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

The consecutive angles of a trapezium form an arithmetic sequence. If the smallest angle is $\displaystyle 75^{\circ}$, then the largest angle is

  1. $\displaystyle 100^{\circ}$
  2. $\displaystyle 105^{\circ}$
  3. $\displaystyle 110^{\circ}$
  4. $\displaystyle 115^{\circ}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since, sum of all the four angles of a quadrilateral is $360^o$.

Angle 1 $=75^o$, Angle 2 $=75^o+x$, Angle 3 $=75^o+2x$, Angle 4 $=75^o+3x$
Angle 1 $+$ Angle 2 $+$ Angle 3 $+$ Angle 4 $=360^o$
$\therefore   75^o+75^o+x+75^o+2x+75^o+3x=360^o$
$\Rightarrow 300+6x=360\Rightarrow 6x=60 \Rightarrow x=10$
$\therefore$ Largest angle (Angle 4)$=75^o+3x=75^o+3\times 10=105^o$

Hence, option B.

Multiple choice maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

In isoceles trapezoid ABCD, side CD is parallel to to side AB, line segment AC is congruent to line segment BD.The degree measure of angle BDC = $80^o$. Find the measures of the $\angle A$.

  1. $90^o$
  2. $100^o$
  3. $110^o$
  4. $120^o$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

As per the property of isosceles trapezoid, 
Opposite sides of an isosceles trapezoid are the same length (congruent) and the angles on either side of the bases are the same size (congruent).
So, Angle C = $80^o$
Since the top and bottom angles are supplementary, we know that,
Angle A = $180 - 80$
Angle A = $100^o$
Similarly, the Angle of B = $100^o$

Multiple choice

What is the sum of the interior angles of a quadrilateral?

  1. 180 degrees

  2. 270 degrees

  3. 360 degrees

  4. 540 degrees
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The sum of the interior angles of any quadrilateral is always 360 degrees.

Multiple choice

What is the name of the theorem that states that the sum of the interior angles of a triangle is equal to 180 degrees?

  1. Brahmagupta's Theorem

  2. Pythagorean Theorem

  3. Euler's Formula

  4. Descartes' Theorem
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Euler's Formula states that the sum of the interior angles of a triangle is equal to 180 degrees.

Multiple choice

What is the name of the theorem that states that the sum of the interior angles of a triangle is always 180 degrees?

  1. Pythagorean theorem

  2. Euclid's theorem

  3. Brahmagupta's theorem

  4. Bhaskara's theorem
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Euclid's theorem, also known as the angle sum property of a triangle, states that the sum of the interior angles of a triangle is always 180 degrees.

Multiple choice

What is the name of the theorem that states that the sum of the angles of a triangle is equal to 180 degrees?

  1. Pythagorean Theorem

  2. Triangle Inequality Theorem

  3. Angle Sum Theorem

  4. Law of Cosines
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The Angle Sum Theorem states that the sum of the angles of a triangle is equal to 180 degrees. This theorem is attributed to Indian mathematicians.

Multiple choice

What is the name of the mathematical theorem that states that the sum of the interior angles of a triangle is always 180 degrees?

  1. Pythagorean Theorem

  2. Triangle Inequality Theorem

  3. Law of Cosines

  4. Angle Addition Postulate
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The Angle Addition Postulate states that the sum of the interior angles of a triangle is always 180 degrees.

Multiple choice

In a right triangle, the angle opposite the hypotenuse is called the:

  1. Adjacent angle

  2. Opposite angle

  3. Hypotenuse angle

  4. Complementary angle
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a right triangle, the angle opposite the hypotenuse is known as the opposite angle.

Multiple choice

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the:

  1. Adjacent and opposite sides

  2. Opposite and hypotenuse sides

  3. Adjacent and hypotenuse sides

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

The Pythagorean theorem relates the squares of the sides of a right triangle.

Multiple choice

The sine of an angle is defined as the ratio of the:

  1. Opposite side to the adjacent side

  2. Opposite side to the hypotenuse

  3. Adjacent side to the hypotenuse

  4. Hypotenuse to the opposite side
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The sine of an angle is calculated by dividing the opposite side by the hypotenuse.

Multiple choice

The cosine of an angle is defined as the ratio of the:

  1. Adjacent side to the hypotenuse

  2. Opposite side to the hypotenuse

  3. Adjacent side to the opposite side

  4. Hypotenuse to the adjacent side
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The cosine of an angle is calculated by dividing the adjacent side by the hypotenuse.

Multiple choice

In a right triangle, the side opposite the right angle is called the:

  1. Hypotenuse

  2. Adjacent side

  3. Opposite side
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The side opposite the right angle in a right triangle is called the opposite side.