CBSE Class 8 Mathematics Quadrilaterals Very Short Answer Questions with Answers

Very Short Answer Questions

Question 1. Write the definition of a rectangle as it is finally stated in this chapter.
Answer:
A rectangle is a quadrilateral in which all the angles are 90°. Equal opposite sides then follow automatically.

Teacher's Note:
1. One condition is enough for the definition.
2. Other properties are deduced from it.

Question 2. From which two Latin words is the word “quadrilateral” formed, and what does each of them mean?
Answer:
From quadri, meaning four, and latus, which refers to sides.

Teacher's Note:
1. The name itself says “four sides”.
2. Compare “lateral”, which also means side.

Question 3. What is the sum of the four angles of any quadrilateral?
Answer:
The four angles of any quadrilateral add up to 360°.

Teacher's Note:
1. A diagonal splits it into two triangles of 180° each.
2. This holds for every quadrilateral, regular or not.

Question 4. Define a square.
Answer:
A square is a quadrilateral in which all the angles are 90° and all the sides are of equal length.

Teacher's Note:
1. A square needs both conditions.
2. So it is both a rectangle and a rhombus.

Question 5. Name the quadrilateral in which all four sides are of the same length.
Answer:
It is a rhombus. (A square is a special rhombus.)

Teacher's Note:
1. Equal sides alone do not fix the angles.
2. Only when the angles are 90° is it a square.

Question 6. State the property of the diagonals of a rectangle.
Answer:
The diagonals of a rectangle are equal in length and bisect each other (they cross at their midpoints).

Teacher's Note:
1. Two facts: equal, and bisecting.
2. They need not be perpendicular.

Question 7. Define a parallelogram.
Answer:
A parallelogram is a quadrilateral in which the opposite sides are parallel.

Teacher's Note:
1. Both pairs of opposite sides must be parallel.
2. Rectangles and rhombuses are special parallelograms.

Question 8. Define a trapezium.
Answer:
A trapezium is a quadrilateral having at least one pair of parallel opposite sides.

Teacher's Note:
1. “At least one” allows two pairs as well.
2. So a parallelogram is also a trapezium.

Question 9. A kite is labelled ABCD. Write the two equalities between its sides that make it a kite.
Answer:
AB = BC and CD = DA, that is, two adjacent pairs of equal sides.

Teacher's Note:
1. The equal sides are next to each other, not opposite.
2. The two pairs do not share a side.

Question 10. What is an isosceles trapezium?
Answer:
It is a trapezium in which the two non-parallel sides are of equal length.

Teacher's Note:
1. Like an isosceles triangle, it has two equal “legs”.
2. Its base angles turn out to be equal.

Question 11. At what angle do the two diagonals of a rhombus cut each other?
Answer:
They cut each other at 90°, and they also bisect each other.

Teacher's Note:
1. Rhombus diagonals are perpendicular bisectors.
2. They need not be equal in length.

Question 12. Is every square a rhombus? Give a reason for your answer.
Answer:
Yes. A rhombus is a quadrilateral with all four sides equal, and every square has all four sides equal.

Teacher's Note:
1. Check the square against the rhombus definition.
2. The reverse is not true.

Question 13. Three angles of a quadrilateral measure 70°, 80° and 100°. Find the fourth angle.
Answer:
Fourth angle = 360° − (70° + 80° + 100°) = 360° − 250° = 110°.

Teacher's Note:
1. Use the angle sum 360°.
2. Check: 70 + 80 + 100 + 110 = 360.

Question 14. In a parallelogram ABCD, ∠A = 65°. Find ∠B and ∠C.
Answer:
Adjacent angles add to 180°: ∠B = 180° − 65° = 115°.
Opposite angles are equal: ∠C = ∠A = 65°.

Teacher's Note:
1. B is next to A; C is opposite A.
2. Adjacent: add to 180°; opposite: equal.

Question 15. In a rhombus PQRS, ∠P = 110°. Find ∠Q and ∠R.
Answer:
A rhombus is a parallelogram, so ∠Q = 180° − 110° = 70° and ∠R = ∠P = 110°.

Teacher's Note:
1. Parallelogram angle rules apply to every rhombus.
2. Opposite angles are equal.

Question 16. The diagonals of a quadrilateral are equal in length and bisect each other. Name the quadrilateral.
Answer:
It is a rectangle; equal diagonals that bisect each other always make all four angles 90°.

Teacher's Note:
1. This is the carpenter's test for a rectangle.
2. The angle between the diagonals can be anything.

Question 17. The diagonals of a quadrilateral bisect each other at right angles, but the two diagonals are of unequal length. Name the quadrilateral.
Answer:
It is a rhombus. Perpendicular bisecting diagonals give four equal sides, and unequal diagonals mean it is not a square.

Teacher's Note:
1. If the diagonals were also equal, it would be a square.
2. Unequal diagonals rule out right angles.

Question 18. In a trapezium PQRS with PQ ∥ SR, ∠P = 115°. Find ∠S.
Answer:
∠P and ∠S are co-interior angles on the transversal PS, so ∠S = 180° − 115° = 65°.

Teacher's Note:
1. Pair the angles at the two ends of a slanting side.
2. Co-interior angles add up to 180°.

Question 19. Two identical equilateral triangles of side 5 cm are joined along a common side. Name the quadrilateral formed and write its four angles.
Answer:
It is a rhombus of side 5 cm, with angles 60°, 120°, 60° and 120°.

Teacher's Note:
1. At the ends of the joined side, 60° + 60° = 120°.
2. The other two corners stay 60°.

Question 20. Can a quadrilateral have three right angles and a fourth angle of 80°? Give a reason.
Answer:
No. The angle sum would be 90° + 90° + 90° + 80° = 350°, but the angles of a quadrilateral must add up to 360°.

Teacher's Note:
1. Three right angles force the fourth to be 90°.
2. Always test with the angle sum.

Question 21. In a square ABCD, the diagonal AC is drawn. Find ∠BAC.
Answer:
The diagonals of a square bisect its angles, so ∠BAC = \( \frac{90^\circ}{2} \) = 45°.

Teacher's Note:
1. △ABC is an isosceles right triangle.
2. Its base angles are 45° each.

Question 22. Why is every rhombus also a parallelogram? Give the reason used in the chapter.
Answer:
A diagonal of a rhombus makes four equal angles with the sides. These equal alternate angles show that both pairs of opposite sides are parallel, so a rhombus is a parallelogram.

Teacher's Note:
1. The isosceles triangles give the equal angles.
2. Equal alternate angles mean parallel lines.

Question 23. By the definition of a trapezium used in this chapter, is every parallelogram a trapezium? Give a reason.
Answer:
Yes. A trapezium needs at least one pair of parallel opposite sides, and a parallelogram has two such pairs.

Teacher's Note:
1. The words “at least one” decide this.
2. Some books use “exactly one”, but not this chapter.

Question 24. In a rhombus ABCD, the diagonal BD makes an angle ∠ABD = 35° with the side AB. Find ∠ABC and ∠BAD.
Answer:
BD bisects ∠B, so ∠ABC = 2 × 35° = 70°.
Adjacent angles add to 180°, so ∠BAD = 180° − 70° = 110°.

Teacher's Note:
1. Double the half-angle first.
2. Then use the adjacent-angle rule.

Question 25. A quadrilateral has four equal sides and one of its angles is 90°. Which quadrilateral is it? Justify in one line.
Answer:
It is a square: it is a rhombus, so the angles next to the 90° angle are 180° − 90° = 90° and the opposite angle is 90°, making all four angles 90°.

Teacher's Note:
1. One right angle in a rhombus makes all four right angles.
2. Equal sides plus right angles define a square.

Question 26. Is the statement “a quadrilateral whose diagonals are perpendicular to each other must be a rhombus” true? Justify your answer.
Answer:
False. A kite also has perpendicular diagonals, yet its four sides are not all equal, so it need not be a rhombus.

Teacher's Note:
1. A single counter-example disproves a “must” statement.
2. The kite is the usual counter-example.

Question 27. On a geoboard, two rubber bands of equal length are stretched so that they bisect each other at right angles. Name the quadrilateral formed by joining their four ends, and say what it becomes if only one band is made longer.
Answer:
It is a square, since the diagonals are equal, bisect each other and meet at 90°.
If one band is lengthened, the diagonals stay perpendicular bisectors but become unequal, so it becomes a rhombus.

Teacher's Note:
1. The bands act as diagonals.
2. Unequal perpendicular bisecting diagonals give a rhombus.

Question 28. In a kite ABCD with AB = BC and CD = DA, ∠ABC = 80° and ∠ADC = 50°. Find ∠BAD.
Answer:
△ABD ≅ △CBD (SSS), so ∠BAD = ∠BCD.
∠BAD + ∠BCD = 360° − 80° − 50° = 230°, so ∠BAD = 115°.

Teacher's Note:
1. The two equal angles of a kite lie between unequal sides.
2. Halve the remaining 230°.

Question 29. CASE is a square and U, V, W, X are the midpoints of its four sides, taken in order. What type of quadrilateral is UVWX?
Answer:
UVWX is a square. The four corner triangles are congruent isosceles right triangles, so its sides are equal, and the two 45° angles at each midpoint leave 180° − 90° = 90° for each angle of UVWX.

Teacher's Note:
1. Each corner triangle has legs equal to half a side.
2. 45° + 90° + 45° = 180° along a side.

Question 30. Two identical triangles of sides 6 cm, 9 cm and 12 cm are joined as mirror images along their 12 cm sides. Name the quadrilateral formed and state one property of its 12 cm diagonal.
Answer:
It is a kite with sides 6 cm, 9 cm, 9 cm and 6 cm.
Its 12 cm diagonal bisects the other diagonal at right angles (it also bisects the angles at its two ends).

Teacher's Note:
1. The joined side becomes the axis of symmetry.
2. Mirror images keep equal sides adjacent.