Class 10 Maths Chapter 6 - Triangles | INSTALEARN

Question 1

Assertion (A): If two triangles are similar, their corresponding angles are equal.
Reason (R): Similar triangles have proportional sides and equal corresponding angles.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Similar triangles are defined by having both proportional sides and equal corresponding angles.

Question 2

Assertion (A): In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Reason (R): This is known as the Pythagorean Theorem.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The statement is the Pythagorean Theorem, which is applicable to right-angled triangles.

Question 3

Assertion (A): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
Reason (R): This is the Side-Angle-Side (SAS) criterion for congruence of triangles.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The SAS criterion is used to establish the congruence of two triangles based on two sides and the included angle.

Question 4

Assertion (A): If two triangles are congruent, they are also similar.
Reason (R): Congruent triangles have equal corresponding sides and angles.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Congruent triangles are indeed similar as they have equal corresponding sides and angles, which meets the definition of similarity.

Question 5

Assertion (A): In a triangle, the angle opposite to the longest side is always the largest angle.
Reason (R): In any triangle, angles opposite longer sides are larger than angles opposite shorter sides.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: This is a fundamental property of triangles.

Question 6

Assertion (A): If the corresponding sides of two triangles are in proportion, then the triangles are similar.
Reason (R): The Side-Side-Side (SSS) criterion is used for proving similarity in triangles.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The SSS criterion proves similarity when the corresponding sides of two triangles are proportional.

Question 7

Assertion (A): The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
Reason (R): This property is known as the Triangle Inequality Theorem.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the third side.

Question 8

Assertion (A): In an isosceles triangle, the base angles are always equal.
Reason (R): The angles opposite equal sides are always equal.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: In an isosceles triangle, two sides are equal, and the angles opposite them are also equal.

Question 9

Assertion (A): The area of a triangle can be calculated if the lengths of all three sides are known.
Reason (R): Heron’s formula is used to find the area of a triangle with given side lengths.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Heron’s formula requires the lengths of all three sides to calculate the area.

Question 10

Assertion (A): Two equilateral triangles with equal areas are congruent.
Reason (R): Equilateral triangles with the same area must have the same side length.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Since equilateral triangles with the same side lengths have equal areas, two equilateral triangles with equal areas must be congruent.

Question 11

Assertion (A): In any triangle, the centroid divides each median in the ratio 2:1.
Reason (R): The centroid is the intersection point of the medians of a triangle.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The centroid divides each median in the ratio 2:1, which is a known property of medians in triangles.

Question 12

Assertion (A): In a triangle, the orthocenter is always inside the triangle.
Reason (R): The orthocenter is the point of intersection of the altitudes of a triangle.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is false, but R is true.
D) A is true, but R is false.
Answer: C) A is false, but R is true.
Explanation: The orthocenter may lie inside, outside, or on the triangle, depending on whether the triangle is acute, obtuse, or right-angled.

Question 13

Assertion (A): The circumcenter of a right-angled triangle lies on the hypotenuse.
Reason (R): The circumcenter is equidistant from all vertices of a triangle.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: B) Both A and R are true, but R is not the correct explanation of A.
Explanation: In a right-angled triangle, the circumcenter lies at the midpoint of the hypotenuse.

Question 14

Assertion (A): If two triangles have equal corresponding sides, then they are congruent.
Reason (R): The Side-Side-Side (SSS) criterion proves the congruence of triangles.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The SSS criterion confirms the congruence of triangles based on equal corresponding sides.

Question 15

Assertion (A): The sum of the interior angles of a triangle is always 180 degrees.
Reason (R): A triangle has three interior angles.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: B) Both A and R are true, but R is not the correct explanation of A.
Explanation: The fact that a triangle has three angles does not explain why their sum is 180 degrees; this is a property of Euclidean geometry.

Question 16

Assertion (A): Two isosceles triangles with equal base lengths are similar.
Reason (R): Isosceles triangles have two equal sides.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: C) A is true, but R is false.
Explanation: Isosceles triangles with equal bases are not necessarily similar; their angles must also correspond.

Question 17

Assertion (A): An equilateral triangle is always similar to another equilateral triangle.
Reason (R): All equilateral triangles have equal angles and proportional sides.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Since equilateral triangles have equal corresponding angles and proportional sides, they are similar.

Question 18

Assertion (A): In any right triangle, the length of the hypotenuse is the longest side.
Reason (R): The hypotenuse is opposite the right angle, which is the largest angle in the triangle.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The side opposite the largest angle (90°) is the longest in a right triangle, hence the hypotenuse.

Question 19

Assertion (A): In any triangle, the longest side is opposite the largest angle.
Reason (R): The larger the angle, the longer the side opposite to it.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: This is a fundamental property of triangles; larger angles are opposite longer sides.

Question 20

Assertion (A): If a triangle has two equal sides, it is an isosceles triangle.
Reason (R): An isosceles triangle has two equal sides and two equal angles.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: B) Both A and R are true, but R is not the correct explanation of A.
Explanation: The definition of an isosceles triangle is having two equal sides; the equal angles are a consequence, not an explanation.

Question 21

Assertion (A): In a right triangle, the sum of the squares of the two shorter sides equals the square of the hypotenuse.
Reason (R): The Pythagorean Theorem applies to all triangles.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: C) A is true, but R is false.
Explanation: The Pythagorean Theorem only applies to right triangles, not all triangles.

Question 22

Assertion (A): If the corresponding angles of two triangles are equal, then the triangles are similar.
Reason (R): Similar triangles have equal angles and proportional sides.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The Angle-Angle (AA) criterion states that if two triangles have equal corresponding angles, they are similar.

Question 23

Assertion (A): If two triangles have the same area, then they are congruent.
Reason (R): Triangles with the same area can be different in shape and size.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: Triangles can have the same area but differ in shape and size, so they are not necessarily congruent.

Question 24

Assertion (A): The altitude of a triangle bisects the opposite side.
Reason (R): An altitude divides the triangle into two right-angled triangles.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: The altitude only bisects the opposite side if it’s an isosceles triangle.

Question 25

Assertion (A): In a triangle, the sum of any two sides is greater than the third side.
Reason (R): This is called the Triangle Inequality Theorem.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

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Class 10 Maths Chapter 6 - Triangles