Angular speed of the planets

No, planets in the solar system do not have the same angular speed.

Firstly, strictly speaking, the angular speeds of the planets change with time because they travel along ellipses, not circles. But the eccentricities of many of the planets are close to $0$ (see e.g. Why are the orbits of planets in the Solar System nearly circular?), so to a reasonable approximation, we can approximate them as traveling along circles to get an idea of their orbital angular velocities.

Kepler's Third Law tells us that the square of the orbital period of a planet is proportional to the cube of its orbit's semi-major axis; begin{align} T^2propto a^3 end{align} Since the semi-major axes of the orbits of the different planets are different, so are their orbital periods, and consequently their angular velocities are different as well. In particular, the Third Law shows that for each planet, begin{align} T_mathrm{planet} = left(frac{a_{mathrm{planet}}}{a_mathrm{Earth}}right)^{3/2}T_mathrm{Earth} end{align} This allows us to determine how much longer and/or shorter the orbital periods of certain planets are, and how much smaller and/or larger their corresponding angular velocities are.

For example, begin{align} a_mathrm{Earth} &approx 15.0times 10^{10},mathrm{m} a_mathrm{Mars} &approx 22.8times 10^{10},mathrm{m} end{align} so begin{align} T_mathrm{Mars} approx left(frac{22.8times 10^{10},mathrm{m}}{15.0times 10^{10},mathrm{m}}right)^{3/2}(1,mathrm{yr}) approx 1.87,mathrm{yr} end{align} and consequently, since the angular velocity $omega$ and orbital period $T$ are related by $T=2pi/omega$, the angular velocity of Earth is roughly 1.87 times the angular velocity of Mars.

You can find a table of more info on this stuff at the end the of this hyperphysics page.

I would guess there is something missing in this question. Are you sure you have the teacher's statement correct? If the planets all had the the same angular speed (about the Sun) they would all complete an orbit in the same amount of time.

If it is angular speed about their axis, again clearly not true. Mercury rotates once in a Mercury year and a Mars day (what they call a Sol in the Mars mission groups - I wonder what they will call it on other planets?) is a little longer than an Earth day.

Plus the planets are in elliptical orbits and their angular velocity changes with their distance from the sun - the equal times, equal area rule of Kepler. In fact, Kepler's three laws clearly state the relation between the time to orbit for bodies at two different distances from the Sun.