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5. BD = 8 cm

6. See explanation below.

7. x = 42; y = 96; z = 64

Step-by-step explanation:

5.

DC = 5 cm

The diagonals of a rhombus bisect each other, so since AC = 6 cm, OC = 3 cm.

The diagonals of a rhombus are perpendicular to each other, so triangle DOC is a right triangle with right angle DOC.

a^2 + b^2 = c^2

(DO)^2 + (OC)^2 = (DC)^2

(DO)^2 + (3 cm)^2 = (5 cm)^2

(DO)^2 + 9 cm^2 = 25 cm^2

(DO)^2 = 16 cm^2

DO = 4 cm

Since BD = 2DO,

BD = 2(4 cm) = 8 cm

BD = 8 cm

6.

There are different definitions of trapezium. In the U.S., trapezium is a quadrilateral, none of whose sides are parallel. According to the U.S. definition of trapezium, then, no parallelogram is a trapezium.

According to the UK definition of trapezium, a trapezium is a quadrilateral with at least 2 parallel sides. That means the other two sides may or may not be parallel. According to this definition, then a parallelogram is always a trapezium.

8.

Triangle ADB is isosceles with AD = AB. That meakes their opposite angles congruent.

x = m<ADB = 42

42 + 42 + y = 180

y = 96

In a kite, there are two pairs of congruent sides. DC = BC, so z = m<BDC

z + z + 52 = 180

2z = 128

z = 64

m∠5 + m∠3 = m∠1   is true.

m∠5 + m∠6 = 180 is true

m∠2 +m∠3 = m∠6   is true

m∠2 +m∠3 +m∠5 = 180   is true