CBSE Class 8 Mathematics Quadrilaterals MCQs Set 11

Answer: (b) sides

Explanation:
1. “Quadrilateral” is made from the Latin quadri (four) and latus (side).
2. So latus refers to sides, and a quadrilateral is a closed figure with four sides.

Teacher's Note:
1. Think of “lateral”, which also means “of the side”.
2. Quadri = four, latus = side: four-sided figure.

Answer: (c) all the angles are 90°

Explanation:
1. At first a rectangle is described by right angles together with equal opposite sides.
2. Deduction 4 shows that equal opposite sides follow on their own once all angles are 90°.
3. So the final, shortest definition is: a quadrilateral in which all the angles are 90°.

Teacher's Note:
1. A good definition uses the fewest conditions needed.
2. All other rectangle properties can be deduced from the right angles.

Answer: (a) 360°

Explanation:
1. A diagonal divides the quadrilateral into two triangles.
2. The angles of each triangle add up to 180°, and together they make up the four angles of the quadrilateral.
3. So the sum is 180° + 180° = 360°.

Teacher's Note:
1. Two triangles make one quadrilateral.
2. 180° is the angle sum of a triangle, not of a quadrilateral.

Answer: (d) rhombus

Explanation:
1. A quadrilateral with all sides of the same length is a rhombus.
2. A kite needs only two adjacent pairs of equal sides, and a rectangle needs only right angles.

Teacher's Note:
1. Rhombus = four equal sides; the angles may be anything.
2. A square is a special rhombus with right angles.

Answer: (c) 90°

Explanation:
1. In square ABCD with diagonals meeting at O, △BOA ≅ △BOC by SSS.
2. So ∠BOA = ∠BOC, and together they form a straight angle of 180°.
3. Hence each is 90°.

Teacher's Note:
1. 45° is the angle between a diagonal and a side, not between the diagonals.
2. Equal angles on a straight line are right angles.

Answer: (d) having at least one pair of parallel opposite sides

Explanation:
1. The chapter defines a trapezium as a quadrilateral having at least one pair of parallel opposite sides.
2. The other options describe a rhombus, a rectangle-type figure and a rectangle, not a trapezium.

Teacher's Note:
1. Note the words “at least one”: a parallelogram also counts as a trapezium.
2. A trapezium need not have any equal sides.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. A is true: every square is a rectangle.
2. R is true: it states the definition of a rectangle and notes that a square has all angles 90°.
3. A square therefore satisfies the definition of a rectangle, so R correctly explains A.

Teacher's Note:
1. To show a shape belongs to a family, check it against the family's definition.
2. The converse is false: a rectangle need not be a square.

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. A is true: with AB as transversal, ∠A + ∠B = 90° + 90° = 180°, so AD ∥ BC; similarly AB ∥ DC.
2. R is true: the diagonals of a rectangle are equal.
3. But the parallel sides come from the right angles, not from the diagonals, so R does not explain A.

Teacher's Note:
1. Both statements can be true and still be unrelated.
2. Ask: does R give the reason for A?

Answer: (c) Both statements are true

Explanation:
1. Two adjacent angles are co-interior angles between a pair of parallel sides, so they add up to 180°. Statement II is true.
2. If ∠A = x, then ∠B = 180° − x and ∠C = 180° − (180° − x) = x, so opposite angles are equal. Statement I is true.
3. So both statements are true.

Teacher's Note:
1. Statement II is the reason behind Statement I.
2. Adjacent: supplementary; opposite: equal.

Answer: (b) 70°

Explanation:
1. The angles of a quadrilateral add up to 360°.
2. Sum of the given angles = 85° + 95° + 110° = 290°.
3. Fourth angle = 360° − 290° = 70°.

Teacher's Note:
1. Add the known angles first, then subtract from 360°.
2. Check: 85 + 95 + 110 + 70 = 360.

Answer: (a) (i)-(r), (ii)-(p), (iii)-(q), (iv)-(s)

Explanation:
1. Rectangle: all four angles are 90°, so (i)-(r).
2. Rhombus: all four sides are equal, so (ii)-(p).
3. Trapezium: at least one pair of parallel opposite sides, so (iii)-(q).
4. Square: all angles 90° and all sides equal, so (iv)-(s). The answer is (a).

Teacher's Note:
1. Match the definition, not just a property.
2. Options (a) and (d) differ only in (ii) and (iii).

Answer: (c) 110°

Explanation:
1. In a parallelogram the opposite angles are equal.
2. ∠C is opposite ∠A, so ∠C = ∠A = 110°.
3. The adjacent angles ∠B and ∠D are 180° − 110° = 70° each.

Teacher's Note:
1. Opposite angles of a parallelogram are equal.
2. 70° is the adjacent angle, a common wrong choice.

Answer: (d) The diagonals are equal in length.

Explanation:
1. Every parallelogram has equal opposite sides, equal opposite angles and diagonals that bisect each other.
2. The diagonals are equal only in a rectangle; equal diagonals that bisect each other force all angles to be 90°.
3. So “the diagonals are equal in length” is not true for every parallelogram.

Teacher's Note:
1. Read “not” carefully in the question.
2. Equal diagonals are a rectangle property.

Answer: (b) bisect each other at right angles

Explanation:
1. A rhombus is a parallelogram, so its diagonals bisect each other.
2. Congruent triangles show that the angles at the meeting point are equal and form a straight angle, so each is 90°.
3. So the diagonals bisect each other at right angles; they need not be equal.

Teacher's Note:
1. Rhombus diagonals: perpendicular bisectors of each other.
2. Equal diagonals happen only when the rhombus is a square.

Answer: (d) BD bisects AC at right angles.

Explanation:
1. BA = BC and DA = DC, so △ABD ≅ △CBD (SSS) and BD bisects ∠B.
2. Then △AOB ≅ △COB (SAS), so AO = OC and the equal angles at O form a straight angle, each 90°.
3. So BD bisects AC at right angles.

Teacher's Note:
1. The diagonal joining the vertices where equal sides meet is the axis of symmetry.
2. That diagonal cuts the other one in half at 90°.

Answer: (c) A is true, but R is false.

Explanation:
1. A is true: △ABD ≅ △CBD by SSS, so ∠ABD = ∠CBD.
2. R is false: the diagonals of a kite are generally of different lengths.

Teacher's Note:
1. Look at a paper kite: one stick is usually longer than the other.
2. SSS congruence is the key to kite properties.

Answer: (d) A is false, but R is true.

Explanation:
1. A is false: a kite also has perpendicular diagonals, but its four sides need not be equal.
2. R is true: the diagonals of a rhombus do bisect each other at right angles.

Teacher's Note:
1. One counter-example is enough to prove a statement false.
2. Perpendicular diagonals alone do not make a rhombus.

Answer: (a) Only Statement I is true

Explanation:
1. A diagonal of a rhombus makes equal alternate angles with the sides, so both pairs of opposite sides are parallel. Statement I is true.
2. A parallelogram need not have all four sides equal (for example 7 cm and 12 cm sides). Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. “Every A is a B” does not mean “every B is an A”.
2. Use one example to reject Statement II.

Answer: (b) Only Statement II is true

Explanation:
1. The equal sides of an isosceles trapezium are its non-parallel sides, so it has only one pair of parallel sides. Statement I is false.
2. Dropping perpendiculars gives two congruent right triangles at the ends, so the base angles are equal. Statement II is true.
3. So only Statement II is true.

Teacher's Note:
1. Equal sides are not the same as parallel sides.
2. Isosceles trapezium: equal base angles.

Answer: (c) 100°

Explanation:
1. ∠P and ∠Q are adjacent angles, so 4x + 5x = 180°, giving 9x = 180° and x = 20°.
2. ∠P = 80° and ∠Q = 100°.
3. ∠S is opposite ∠Q, so ∠S = 100°.

Teacher's Note:
1. Find x first, then the angles.
2. 20° is x, not an angle of the parallelogram.

Answer: (a) 60°

Explanation:
1. Total parts = 2 + 3 + 3 + 4 = 12, and the angles add up to 360°.
2. One part = 360° ÷ 12 = 30°.
3. Smallest angle = 2 × 30° = 60° (the angles are 60°, 90°, 90°, 120°).

Teacher's Note:
1. Divide the total angle by the total number of parts.
2. Check: 60 + 90 + 90 + 120 = 360.

Answer: (d) 55°

Explanation:
1. Adjacent angles of a rhombus add up to 180°, so ∠B = 180° − 70° = 110°.
2. The diagonal BD bisects ∠B, so ∠ABD = \( \frac{110^\circ}{2} \) = 55°.

Teacher's Note:
1. 35° is half of ∠A, but BD bisects ∠B, not ∠A.
2. Check in △ABD: 70° + 55° + 55° = 180°.

Answer: (c) (i)-(r), (ii)-(p), (iii)-(s), (iv)-(q)

Explanation:
1. Equal diagonals bisecting each other at 90°: square, (i)-(r).
2. Diagonals that only bisect each other: parallelogram, (ii)-(p).
3. Diagonals bisecting at 90° but not equal: rhombus, (iii)-(s).
4. One diagonal bisecting the other at 90°: kite, (iv)-(q). The answer is (c).

Teacher's Note:
1. The more conditions, the more special the shape.
2. Kite is the only one where just one diagonal is bisected.

Answer: (b) 75°

Explanation:
1. PQ ∥ SR and PS is a transversal, so ∠P and ∠S are interior angles on the same side.
2. ∠S = 180° − 105° = 75°.
3. Similarly ∠Q = 180° − 120° = 60°; check: 105 + 60 + 120 + 75 = 360.

Teacher's Note:
1. Pair each angle with the angle on the same slanting side.
2. ∠S pairs with ∠P, not with ∠R.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. R is true: if each angle is x, then 4x = 360° and x = 90°.
2. A is true: a quadrilateral with all angles 90° is a rectangle by definition.
3. R gives exactly the reason why A holds, so R correctly explains A.

Teacher's Note:
1. Use the angle sum to turn “equal” into “90° each”.
2. Then apply the definition of a rectangle.

Answer: (c) A is true, but R is false.

Explanation:
1. A is true: bisecting diagonals give congruent triangles (SAS), whose equal alternate angles make both pairs of opposite sides parallel.
2. R is false: the diagonals of a general parallelogram are not perpendicular; that happens only in a rhombus.

Teacher's Note:
1. Bisecting diagonals is a test for a parallelogram.
2. Perpendicular diagonals belong to a rhombus or kite.

Answer: (a) Only Statement I is true

Explanation:
1. The 12 cm side is hidden inside; going round the figure the sides are 6 cm, 9 cm, 9 cm, 6 cm.
2. The equal sides come in two adjacent pairs, so it is a kite. Statement I is true.
3. Its opposite sides (6 cm and 9 cm) are unequal, so it is not a parallelogram. Statement II is false.
4. So only Statement I is true.

Teacher's Note:
1. Mirror-image joining keeps equal sides next to each other.
2. A parallelogram must have equal opposite sides.

Answer: (b) 116°

Explanation:
1. The diagonals of a rhombus bisect its angles, so ∠A = 2 × 32° = 64°.
2. Adjacent angles add up to 180°, so ∠ABC = 180° − 64° = 116°.

Teacher's Note:
1. 58° is ∠OBA, half of the answer.
2. Double first, then use adjacent angles.

Answer: (d) (i)-(q), (ii)-(s), (iii)-(p), (iv)-(r)

Explanation:
1. Equal diagonals that bisect each other: rectangle, (i)-(q).
2. Four equal sides with one right angle makes all angles 90°: square, (ii)-(s).
3. Diagonals bisecting each other at right angles: rhombus, (iii)-(p).
4. Exactly one pair of parallel sides: trapezium, (iv)-(r). The answer is (d).

Teacher's Note:
1. A rhombus with one right angle is a square.
2. Equal bisecting diagonals give a rectangle.

Answer: (b) 115°

Explanation:
1. △ABD ≅ △CBD by SSS, so ∠BAD = ∠BCD.
2. Angle sum: ∠BAD + ∠BCD = 360° − 80° − 50° = 230°.
3. So ∠BAD = 230° ÷ 2 = 115°.

Teacher's Note:
1. The two angles between unequal sides of a kite are equal.
2. 230° is the sum of both, not one angle.