Drop altitudes from the endpoints of the shorter base to the longer base. In an isosceles trapezoid, this gives two congruent right triangles.

The difference of the bases is \(10-4=6\), so the projection of each leg onto the longer base is \(\dfrac{6}{2}=3\).

Then the height of the trapezoid is:

\(h=\sqrt{5^2-3^2}=\sqrt{25-9}=4\).

The area of the trapezoid is:

\(S=\dfrac{10+4}{2}\cdot 4=7\cdot 4=28\).

Answer: \(28\text{ cm}^2\).

The area of a rhombus is given by

\(S=\dfrac{d_1d_2}{2}\).

Substitute the data:

\(30=\dfrac{10\cdot d_2}{2}=5d_2\), hence \(d_2=6\).

The diagonals of a rhombus intersect at right angles and bisect each other, so the half-diagonals are \(5\) and \(3\).

The side of the rhombus is the hypotenuse of a right triangle:

\(a=\sqrt{5^2+3^2}=\sqrt{25+9}=\sqrt{34}\).

Answer: \(\sqrt{34}\text{ cm}\).

The difference of the bases is \(12-4=8\). In an isosceles trapezoid, after dropping the altitudes, this difference is split equally, so in each right triangle the horizontal leg is \(4\).

Since one of the acute angles is \(45^\circ\), in the right triangle the height equals the other leg:

\(h=4\).

Now the area is:

\(S=\dfrac{4+12}{2}\cdot 4=8\cdot 4=32\text{ cm}^2\).

Answer: \(32\text{ cm}^2\).

The diagonals of a parallelogram divide it into four triangles of equal area.

So the area of the whole parallelogram is:

\(S=4\cdot 24=96\text{ cm}^2\).

The altitude to the side \(12\) is \(\dfrac{96}{12}=8\text{ cm}\).

The altitude to the side \(8\) is \(\dfrac{96}{8}=12\text{ cm}\).

The smaller altitude is \(8\text{ cm}\).

Answer: \(8\text{ cm}\).

In a right trapezoid, two angles are right angles, and the other two sum to \(180^\circ\). If one of them is \(135^\circ\), then the other acute angle is \(45^\circ\).

The difference of the bases is \(12-8=4\text{ cm}\).

This difference is the horizontal leg of the right triangle near the slanted side. Since the acute angle is \(45^\circ\), the height of the trapezoid equals the other leg:

\(h=4\text{ cm}\).

Therefore,

\(S=\dfrac{12+8}{2}\cdot 4=10\cdot 4=40\text{ cm}^2\).

Answer: \(40\text{ cm}^2\).