The quantities which have the same dimensions as those of solid angle are:

To find the quantities that have the same dimensions as those of a solid angle, let's analyze the dimensions of each quantity mentioned in the problem.

First, let's consider the solid angle ($\Omega$).
The solid angle is defined as the ratio of the area of a portion of a spherical surface to the square of the radius of the sphere:
$\Omega =\frac{\mathrm{Area}}{{\mathrm{Radius}}^2}$
The dimension of area is $L^2$, and the dimension of the square of the radius is also $L^2$.
Therefore, the dimensional formula for solid angle is:
$\left[\Omega \right]=\frac{\left[L^2\right]}{\left[L^2\right]}=\left[M^0{L}^0{T}^0\right]$
Thus, the solid angle is a dimensionless quantity.

Now, let's evaluate the dimensions of the quantities in the options:

1. Strain:
Strain is defined as the ratio of change in dimension to the original dimension (for example, change in length divided by original length):
$\mathrm{Strain}=\frac{\Delta L}{L}$
Since it is the ratio of two identical physical quantities (lengths), it is dimensionless:
$\left[\mathrm{Strain}\right]=\left[M^0{L}^0{T}^0\right]$

2. Angle (Plane Angle):
A plane angle ($\theta$) is defined as the ratio of arc length to the radius of the circle:
$\theta =\frac{\mathrm{Arc}}{\mathrm{Radius}}$
Since both arc length and radius have the dimension of length ($L$), the plane angle is also dimensionless:
$\left[\mathrm{Angle}\right]=\frac{\left[L\right]}{\left[L\right]}=\left[M^0{L}^0{T}^0\right]$

3. Stress:
Stress is defined as force per unit area:
$\mathrm{Stress}=\frac{\mathrm{Force}}{\mathrm{Area}}$
Its dimensional formula is:
$\left[\mathrm{Stress}\right]=\frac{\left[ML{T}^{-2}\right]}{\left[L^2\right]}=\left[M{L}^{-1}{T}^{-2}\right]$
This is not dimensionless.

4. Arc:
An arc is a segment of a curve, which has the dimension of length:
$\left[\mathrm{Arc}\right]=\left[L\right]$
This is not dimensionless.

5. Angular Speed ($\omega$):
Angular speed is the rate of change of angular displacement:
$\omega =\frac{d\theta }{dt}$
Its dimensional formula is:
$\left[\omega \right]=\left[T^{-1}\right]$
This is not dimensionless.

Comparing these results, we find that both strain and angle are dimensionless quantities, meaning they have the dimensional formula $\left[M^0{L}^0{T}^0\right]$, which is identical to that of a solid angle.

Therefore, the correct option is strain and angle.