Exercise 12.2

1. State whether true or false

Answer

(i) All rectangles are squares FalseTrue

(ii) All rhombuses are parallelograms TrueFalse

(iii) All squares are rhombuses and also rectangles TrueFalse

(iv) All squares are not parallelograms FalseTrue

(v) All kites are rhombuses FalseTrue

(vi) All rhombuses are kites TrueFalse

(vii) All parallelograms are trapeziums TrueFalse

(viii) All squares are trapeziums TrueFalse

2. Explain how a square is a (i) quadrilateral (ii) parallelogram (iii) rhombus (iv) rectangle.

Answer

A square is a quadrilateral because it has sides.

A square is a parallelogram because its sides are parallel.

A square is a rhombus because all its sides are in length.

A square is a rectangle because all its angles are angles .

3. In a rhombus ABCD, ∠ABC = 40°. Find the other angles.

Answer

In rhombus ABCD:

∠ABC = ° (Given)

∠ADC = ∠= 40° (Opposite angles of a rhombus are equal)

∠BAD + ∠ = 180° (Adjacent angles of a rhombus are supplementary)

∠BAD + ° = 180°

∠BAD = 180° - 40° = °

∠ = ∠BAD = 140° (Opposite angles of a rhombus are equal)

The angles are: ∠ABC = 40°, ∠ADC = 40°, ∠BAD = 140°, ∠BCD = 140°

4. The adjacent angles of a parallelogram are x° and (2x + 30)°. Find all the angles of the parallelogram.

Answer

In parallelogram PQRS:

Adjacent angles are supplementary, so: ∠P + ∠Q = °

x + (2x + 30) = °

x + 2x + 30 = °

x + 30 = 180

3x = 180 -

3x =

x = °

∠P = ∠R = ° (Opposite angles are equal)

∠Q = ∠S = 2() + 30 = °

The angles are: ∠P = 50°, ∠Q = 130°, ∠R = 50°, ∠S = 130°

5. Explain how DEAR is a trapezium. Which of its two sides are parallel?

Answer

In quadrilateral DEAR:

∠D = ° (Given)

∠A = ° (Given)

∠D + ∠A = ° + ° = °

Since ∠D + ∠A = 180°, they are co-interior angles on the same side of a transversal

Therefore, DE || (One pair of opposite sides are parallel)

DEAR is a trapezium because it has exactly pair of parallel sides

The parallel sides are and

6. BASE is a rectangle. Its diagonals intersect at O. Find x, if OB = 5x+1 and OE = 2x + 4.

Answer

In rectangle BASE:

The diagonals of a rectangle are and bisect each other

Since O is the intersection point, OB = OE (Half diagonals are )

OB = (Given)

OE = (Given)

Setting them equal: 5x + 1 =

5x - 2x = 4 -

x =

x =

7. Is quadrilateral ABCD a parallelogram, if ∠A = 70° and ∠C = 65°? Give reason.

Answer

In quadrilateral ABCD:

∠A = ° (Given)

∠C = ° (Given)

In a parallelogram, opposite angles are

But ∠A ≠ ∠C (70° ≠ °)

Since opposite angles are not equal, ABCD is a parallelogram

8. Two adjacent sides of a parallelogram are in the ratio 5:3. The perimeter of the parallelogram is 48cm. Find the length of each of its sides.

Answer

Let the adjacent sides be 5x and 3x:

Adjacent sides are in ratio :

Let the sides be and

In a parallelogram, opposite sides are

Perimeter = 2(5x + 3x) = cm

2(x) = 48

16x =

x =

First side = 5x = 5 × = cm

Second side = 3x = 3 × = cm

The sides are: 15cm, 9cm, 15cm, 9cm

9. The diagonals of the quadrilateral are perpendicular to each other. Is such a quadrilateral always a rhombus? Draw a rough figure to justify your answer.

Answer

A quadrilateral with perpendicular diagonals is always a rhombus.

For a rhombus, diagonals must be AND bisect each other

A kite also has diagonals but is not a rhombus

In a kite, only diagonal bisects the other

In a rhombus, diagonals bisect each other

Therefore, perpendicular diagonals alone do not guarantee a

10. ABCD is a trapezium in which AB || DC. If ∠A = ∠B = 30°, what are the measures of the other two angles?

Answer

In trapezium ABCD with AB || DC:

∠A = ∠B = ° (Given)

Since AB || DC, co-interior angles are

∠A + ∠D = ° (Co-interior angles)

° + ∠D = 180°

∠D = 180° - ° = °

∠B + ∠C = ° (Co-interior angles)

° + ∠C = 180°

∠C = 180° - ° = °

The angles are: ∠A = 30°, ∠B = 30°, ∠C = 150°, ∠D = 150°

11. Fill in the blanks.

(i) A parallelogram in which two adjacent sides are equal is a .

(ii) A parallelogram in which one angle is 90° and two adjacent sides are equal is a .

(iii) In trapezium ABCD, AB || DC. If ∠D = x° then ∠A = - x°.

(iv) Every diagonal in a parallelogram divides it into triangles.

(v) In parallelogram ABCD, its diagonals AC and BD intersect at O. If AO = 5cm then AC = cm.

(vi) In a rhombus ABCD, its diagonals intersect at 'O'. Then ∠AOB = degrees.

(vii) ABCD is a parallelogram then ∠A - ∠C = degrees.

(viii) In a rectangle ABCD, the diagonal AC = 10cm then the diagonal BD = cm.

(ix) In a square ABCD, the diagonal AC is drawn. Then ∠BAC = degrees.