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Giant stars (or red giants) are stars that are most easily identified as those larger than our sun. This is a matter of perspective, as our star is only a middle sized star but still well above average when compared to a large percentage of stars in the Milky Way. Giant stars become like this when all the hydrogen fuel in an average star's core is used up and it starts to fuse helium as its main fuel, which causes it to rapidly expand during the next phase of its life. Our sun will eventually become a red giant, expanding in size well past Earth.
Supergiant stars (or red supergiants) are stars that are even more luminous than giants with large radii as well (size) and high spectra (K and O class stars) which tell how high their surface temperature is, which is usually 3-4 hotter (up to 20,000 Kelvin) than our Sun. Their large sizes allow them to fuse elements above helium right away, but also shorten their lifespans to usually less than a billion years, due to how much energy they put out, causing their nuclear fuel to be used up more quickly. Rigel is a prime example as well as several Wolf-Rayet stars.
White dwarfs are the end life cycle of low-mass stars (between 1-3 solar masses, where 1 solar mass=our sun) that shed their outer layer when they exhaust their fuel. They have extremely dense gravity , but not as much as a neutron star. Examples include Sirus A, which is among the 8 or so dwarfs within the 100 closest star systems to our Sun.
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