Triangles (Class 10 Mathematics)

Triangles are fundamental geometric shapes. Understanding similarity, congruence, and the properties of various triangle types is essential for geometry an

Triangles (Class 10 Mathematics)

Triangles are fundamental geometric shapes. Understanding similarity, congruence, and the properties of various triangle types is essential for geometry an

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TL;DR: Triangles are fundamental geometric shapes. Understanding similarity, congruence, and the properties of various triangle types is essential for geomet…

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Triangles are fundamental geometric shapes. Understanding similarity, congruence, and the properties of various triangle types is essential for geometry an

Triangles MCQs with Answers & Explanations

Q1. Two triangles are congruent if their corresponding sides and angles are:

  1. Proportional
  2. Perpendicular
  3. Equal ✓ (correct)
  4. Parallel

Explanation: Congruent triangles have exactly equal corresponding sides and angles, meaning they have the same shape and size.

Q2. The AAA criterion ensures triangles are:

  1. Isosceles
  2. Congruent
  3. Right-angled
  4. Similar ✓ (correct)

Explanation: If all three angles of one triangle are equal to all three angles of another (AAA), the triangles are similar but not necessarily congruent.

Q3. In a right-angled triangle, if the two legs are 3 and 4, the hypotenuse is:

  1. 5 ✓ (correct)
  2. 6
  3. √25
  4. 7

Explanation: By Pythagoras' theorem: c² = 3² + 4² = 9 + 16 = 25, so c = 5.

Q4. Two triangles are similar if their corresponding angles are:

  1. Supplementary
  2. Equal ✓ (correct)
  3. Complementary
  4. Proportional

Explanation: Similar triangles have equal corresponding angles, causing their sides to be proportional.

Q5. The sum of angles in any triangle is:

  1. 270°
  2. 360°
  3. 180° ✓ (correct)
  4. 90°

Explanation: The angle sum property states that the sum of all angles in a triangle is always 180°.

Q6. If triangle ABC ~ triangle PQR with a scale factor of 2:3, the ratio of their areas is:

  1. 9:4
  2. 2:3
  3. 8:27
  4. 4:9 ✓ (correct)

Explanation: If linear scale factor is k, then area ratio = k². Here, area ratio = (2/3)² = 4/9.

Q7. In an isosceles triangle, the angles opposite the equal sides are:

  1. Equal ✓ (correct)
  2. Different
  3. Complementary
  4. Supplementary

Explanation: In an isosceles triangle, the base angles (opposite the equal sides) are equal.

Q8. The Pythagorean theorem applies to:

  1. Isosceles triangles
  2. Right-angled triangles only ✓ (correct)
  3. All triangles
  4. Obtuse triangles

Explanation: The Pythagorean theorem (a² + b² = c²) applies specifically to right-angled triangles.

Q9. If a triangle has sides 5, 12, and 13, it is:

  1. Equilateral
  2. Obtuse-angled
  3. Right-angled ✓ (correct)
  4. Acute-angled

Explanation: Since 5² + 12² = 25 + 144 = 169 = 13², this is a right-angled triangle (Pythagorean triple).

Q10. In a triangle, the sum of any two sides must be ____ the third side.

  1. Less than
  2. Equal to
  3. At most
  4. Greater than ✓ (correct)

Explanation: The triangle inequality theorem states that the sum of any two sides must be greater than the third side.

Q11. If two triangles are similar, then their corresponding angles are:

  1. Equal ✓ (correct)
  2. Supplementary
  3. Complementary
  4. Proportional

Explanation: If two triangles are similar, their corresponding angles are equal and corresponding sides are proportional.

Q12. The basic proportionality theorem states that if a line is drawn parallel to one side of a triangle, then it divides the other two sides:

  1. Equally
  2. Proportionally ✓ (correct)
  3. At their midpoints
  4. In ratio 1:1

Explanation: The basic proportionality theorem (Thales' theorem) states that a line parallel to one side divides the other two sides proportionally.

Q13. If in triangle ABC, AB = 3 cm, BC = 4 cm, and AC = 5 cm, then triangle ABC is:

  1. Obtuse-angled
  2. Acute-angled
  3. Right-angled ✓ (correct)
  4. Equilateral

Explanation: Since 3^2 + 4^2 = 9 + 16 = 25 = 5^2, triangle ABC is right-angled at B.

Q14. The area of a triangle with sides a, b, and c is given by:

  1. sqrt(a^2 + b^2 - 2ab cosC)
  2. (1/2) x base x height
  3. a x b x c
  4. sqrt(s(s-a)(s-b)(s-c)) ✓ (correct)

Explanation: Heron's formula gives the area of a triangle as sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2.

Q15. If triangle ABC is similar to triangle PQR, and AB:PQ = 2:3, then the ratio of their areas is:

  1. 4:9 ✓ (correct)
  2. 8:27
  3. 6:9
  4. 2:3

Explanation: The ratio of areas of similar triangles is the square of the ratio of corresponding sides: (2/3)^2 = 4/9.

Q16. Two triangles are congruent if:

  1. They are similar
  2. SSS, SAS, ASA, or AAS conditions are satisfied ✓ (correct)
  3. Their corresponding angles are equal
  4. They have the same area

Explanation: Two triangles are congruent if they satisfy any of the congruence criteria: SSS, SAS, ASA, AAS, or RHS.

Q17. In a right-angled triangle, if the hypotenuse is 13 cm and one side is 5 cm, the other side is:

  1. 8 cm
  2. 9 cm
  3. 12 cm ✓ (correct)
  4. 10 cm

Explanation: Using Pythagorean theorem: other side = sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12 cm.

Q18. The angle bisector theorem states that the angle bisector divides the opposite side in the ratio of:

  1. The altitudes
  2. The other two angles
  3. 1:1
  4. The adjacent sides ✓ (correct)

Explanation: The angle bisector theorem states that the angle bisector divides the opposite side in the ratio of the adjacent sides.

Q19. If the altitude from vertex A to side BC of triangle ABC is 6 cm and BC = 8 cm, the area is:

  1. 24 cm^2 ✓ (correct)
  2. 36 cm^2
  3. 12 cm^2
  4. 48 cm^2

Explanation: Area of triangle = (1/2) x base x height = (1/2) x 8 x 6 = 24 cm^2.

Q20. If triangle ABC has sides 6 cm, 8 cm, and 10 cm, then the radius of its circumcircle is:

  1. 4 cm
  2. 5 cm ✓ (correct)
  3. 3 cm
  4. 6 cm

Explanation: For a right-angled triangle, the circumradius = hypotenuse/2 = 10/2 = 5 cm.

Frequently Asked Questions

What are the criteria for congruence of triangles?

The main congruence criteria are SSS (all sides equal), SAS (two sides and included angle equal), ASA (two angles and included side equal), and RHS (right angle, hypotenuse, and one side equal in right triangles).

How is the similarity of triangles useful in real-world applications?

Similar triangles help in indirect measurement (like finding the height of a building using its shadow), scale modeling, surveying, and map-making. The proportionality of sides allows us to calculate unknown lengths without direct measurement.

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