Three phase power ----- isn't each leg 120 degrees from the others?
yes


Single phase power - 220v - isn't each leg 180 degrees from the other?
Single phase means there is one leg only. Think of single phase as
line segment (one line) and three phase as a triangle (three lines).

A lot of the confusion in power phases is that you need to look at where you
are measuring the voltages and where the ground is.

If you are just connecting the two hot leads of the 240 line into your
house, you have one 240 volt phase.
If you also connect the ground, you have two 120 volt phases that are 180
degrees apart.

Three phase power has the three phases 120 degrees apart but the voltages
and ground can be referenced different ways. You can also refer to them in
different ways, depending on how you are using the power.
The common ones are Y with the voltage measured between each phase and the
ground centered between the phases and Delta where the voltage is measured
between the ends of the phases.

208/120 (Y) is common for large buildings because it gives three 120 volt
phases using the ground to each phase. If you connect directly between two
phases (withoug connecting to the ground) you have a single phase 208 volt
connection. A third option is to connect all three phases (but not the
ground) and have a 208 three phase delta power.

240 Delta with a ground at the mid point of one phase is common in small
industrial buildings. It gives the 240 three phase power that many machines
want , allows you to connect single phase 240 equipment (if it doesn't need
a ground) to any of the phases, connect single phase 240 equipment (if it
needs a ground) to the phase with the ground, and two 120 phases on the
grounded phase.

If the available three phase power is the common 240 V delta with the ground
in a leg, your 240V single phase machine is absolutely going to work when
connected to the leg with the ground.
If the machine does not need the ground as reference, it can work on any 240
volt phase.

If you have 208/120 Y three phase power and the machine is not rated to run
on 240/208, then it becomes messier. If you don't need a ground, you can
run a 208-240 boost transformer (autotransformer that just makes up the
difference between the 208 and 240) to run it.
If you need a mid phase ground, you could run a 208-240 center tapped
transformer. You connect the input to a 208 phase and the output's center
tap to the ground.

Isn't this "fun".