Mathematics ยท 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

What is the name of the trigonometric function that is the ratio of the length of the hypotenuse to the length of the opposite side in a right triangle?

  1. Sine

  2. Cosine

  3. Tangent

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

The trigonometric function that is the ratio of the length of the hypotenuse to the length of the opposite side in a right triangle is called the cosecant.

Multiple choice

What is the name of the trigonometric function that is the ratio of the length of the hypotenuse to the length of the adjacent side in a right triangle?

  1. Sine

  2. Cosine

  3. Tangent

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

The trigonometric function that is the ratio of the length of the hypotenuse to the length of the adjacent side in a right triangle is called the secant.

Multiple choice

What is the name of the trigonometric function that is the ratio of the length of the adjacent side to the length of the opposite side in a right triangle?

  1. Sine

  2. Cosine

  3. Tangent

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

The trigonometric function that is the ratio of the length of the adjacent side to the length of the opposite side in a right triangle is called the cotangent.

Multiple choice

What is the name of the trigonometric function that is the difference between the length of the hypotenuse and the length of the cosine in a right triangle?

  1. Sine

  2. Cosine

  3. Tangent

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

The trigonometric function that is the difference between the length of the hypotenuse and the length of the cosine in a right triangle is called the versine.

Multiple choice

What is the name of the theorem that states that the sum of the squares of the two sides of a right triangle is equal to the square of the hypotenuse?

  1. Pythagorean Theorem

  2. Euler's Theorem

  3. Fermat's Last Theorem

  4. Goldbach's Conjecture
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Pythagorean Theorem is one of the most well-known theorems in mathematics and was known to Indian mathematicians long before Pythagoras.

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. Pythagoras' Theorem

  2. Euler's Formula

  3. Bhaskara's Formula

  4. Angle Sum Theorem
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The Angle Sum Theorem is a mathematical theorem that states that the sum of the angles of a triangle is equal to 180 degrees. It is one of the most basic and well-known theorems in geometry. The Angle Sum Theorem has applications in various fields, including surveying, engineering, and architecture.

Multiple choice

What is the formula for the sine of an angle in a right triangle, as given by Bhaskara II?

  1. $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
  2. $\sin \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
  3. $\sin \theta = \frac{\text{opposite}}{\text{adjacent}}$
  4. $\sin \theta = \frac{\text{hypotenuse}}{\text{adjacent}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Bhaskara II's formula for the sine of an angle is $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$. This formula is still used today in trigonometry.

Multiple choice

What is the formula for the cosine of an angle in a right triangle, as given by Bhaskara II?

  1. $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
  2. $\cos \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
  3. $\cos \theta = \frac{\text{opposite}}{\text{adjacent}}$
  4. $\cos \theta = \frac{\text{hypotenuse}}{\text{adjacent}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Bhaskara II's formula for the cosine of an angle is $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$. This formula is still used today in trigonometry.

Multiple choice

What is the formula for the tangent of an angle in a right triangle, as given by Bhaskara II?

  1. $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
  2. $\tan \theta = \frac{\text{adjacent}}{\text{opposite}}$
  3. $\tan \theta = \frac{\text{hypotenuse}}{\text{opposite}}$
  4. $\tan \theta = \frac{\text{hypotenuse}}{\text{adjacent}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Bhaskara II's formula for the tangent of an angle is $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$. This formula is still used today in trigonometry.

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. Triangle Sum Theorem

  2. Pythagoras' Theorem

  3. Euler's Theorem

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

The Triangle Sum Theorem, also known as the Angle Sum Theorem, states that the sum of the interior angles of a triangle is always 180 degrees.

Multiple choice

Which of the following is an example of a mathematical proof?

  1. The Pythagorean theorem can be proven using the Pythagorean triple (3, 4, 5).

  2. The sum of the interior angles of a triangle is 180 degrees.

  3. The area of a circle is pi times the radius squared.

  4. The derivative of sin(x) is cos(x).
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A mathematical proof is a logical argument that demonstrates the truth of a mathematical statement. The Pythagorean theorem can be proven using the Pythagorean triple (3, 4, 5) because it can be shown that the square of the hypotenuse (5) is equal to the sum of the squares of the other two sides (3 and 4).

Multiple choice

Which of the following is an example of an indirect proof?

  1. The Pythagorean theorem can be proven using the Pythagorean triple (3, 4, 5).

  2. The sum of the interior angles of a triangle is 180 degrees.

  3. The area of a circle is pi times the radius squared.

  4. The derivative of sin(x) is cos(x).
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The sum of the interior angles of a triangle can be proven indirectly by showing that its negation, "The sum of the interior angles of a triangle is not 180 degrees", is false. This can be done by showing that the sum of the interior angles of a triangle is always greater than 180 degrees or less than 180 degrees, which is a contradiction.

Multiple choice

Which of the following is an example of a proof by cases?

  1. The Pythagorean theorem can be proven using the Pythagorean triple (3, 4, 5).

  2. The sum of the interior angles of a triangle is 180 degrees.

  3. The area of a circle is pi times the radius squared.

  4. Every integer is either even or odd.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The statement "Every integer is either even or odd" can be proven by cases by considering all possible cases: an integer is either even or odd. There are no other possibilities.

Multiple choice

Which of the following is an example of a non-constructive proof?

  1. The Pythagorean theorem can be proven using the Pythagorean triple (3, 4, 5).

  2. The sum of the interior angles of a triangle is 180 degrees.

  3. The area of a circle is pi times the radius squared.

  4. There exists a real number between 0 and 1.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The statement "There exists a real number between 0 and 1" can be proven non-constructively by showing that the negation of the statement, "There does not exist a real number between 0 and 1", is false. This can be done by showing that the interval [0, 1] is non-empty.

Multiple choice

Which of the following is an example of a proof by mathematical induction?

  1. The Pythagorean theorem can be proven using the Pythagorean triple (3, 4, 5).

  2. The sum of the interior angles of a triangle is 180 degrees.

  3. The area of a circle is pi times the radius squared.

  4. The sum of the first n natural numbers is n(n+1)/2.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The statement "The sum of the first n natural numbers is n(n+1)/2" can be proven by mathematical induction by showing that it is true for n = 1 and that if it is true for n = k, then it is also true for n = k+1.