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Ghost: Because Kayle is quite slow early game you will want this for two reasons-
1. It's a good escape technique
2. Gets you back to the turret faster

Ignite: Good for killing escaping enemy champs or for initiating. Also good when an enemy is using a health potion because it greatly debuffs the healing effect.

Holy Fervor: Gives you 15% of you AP as AD and 30% of your AD as AP. You will want to either: have a nearly 1:1 ratio of AP:AD or have higher AD (you get more AP for AD then you get AD for AP).

Reckoning: Deals damage and slows, Kayle deals bonus damage for the duration of the slow. this is a good initiator because it slows and gives you extra damage for for a short period of time. For the same reason it is also good for chasing.

Divine Blessing: Heals target and provides bonus movement speed for a short time. This can save you or your allies and can be spammed if you have CDR. You can also use it to improve your (or an ally champion's) speed for a short time.

Righteous Fury: Gives you bonus magic damage and makes your attack ranged. This is your major damage output and farming ability. If you have a good balance of points between your Reckoning and this ability you will have the potential to kill enemy champs.

Intervention: Makes the target immune to all damage. This can greatly effect the outcome of a battle. Make sure not to use it at the beginning and when you have most of your hp as it will be useless and most likely, in the latter, cause you to get focused. Using this on the main carry in team fights completely change the outcome of the battle. This could be the difference of who gets an ace, your team or the enemy team.

You will need to sell the Doran's ring to finish these builds. I recommend waiting until you get the Hextech GB, because then you'll have the lifesteal and spell vamp to stay alive.

Berserker's Greaves are great for Kayle! She benefits from high attack speed, but if you'd rather have some magic penetration feel free to get Sorcerer's Shoes instead. I like the attack speed early so I get these right after my Doran's ring.

Nashor's Tooth is good because it gives you not only attack speed but also AP and CDR.

Hextech Gunblade is a must for Kayle. It is the ultimate hybrid item, giving you massive amounts of AP and AD plus lifesteal and Spell Vamp. Once you get this item you can sell your Doran's ring because it's no longer needed.

The Black Cleaver is good for Kayle because it gives you quite a bit of AD while also giving you even more attack speed. Not often seen in Kayle builds, but her passive makes it quite viable.

Void staff gives you AP while also allowing your magic attacks to penetrate through 40% of their MR, which is (obviously) really great with Righteous fury.

Trinity force may seem like a rather spendy item to buy for a 450 champ....but I like it on Kayle....it gives you bonus damage when you use abilities (deadly on Kayle), critical strike, and a chance to slow on basic attacks. Great item for hybrid champs, Kayle is no different.

Malady is good on Kayle because the passive lowering of the target's magic resist makes Righteous Fury more effective. The bonus magic damage also stacks with it and Madred's Bloodrazor.

Manamune is good because it gives you a lot Mana and AD. This alone makes it a very good item for hybrid champs. It boosts the Archangel staff and vice-versa, so you end up with quite a lot of both AD and AP.

Madred's Bloodrazor is a good item in general for any champion who relies on basic attacks. It adds 4% of the targets Max health to your attacks as magic damage. As previously stated this stacks not only with the bonus damage from Malady but also with Righteous fury (making you a force to be reckoned with). And, of course, more attack speed is ALWAYS good.

Warmog's + Atma's works because Kayle is a pretty good farmer so you can build up a warmog's HP stacks pretty fast (therefore building up AD fast)

Youmuu's GB is nice because its active gives you AS. It also gives you AD, CDR, and Armor pen.

Guinsoo's (ironically) is not my favorite item for Kayle. I hear often that it is a bread-and-butter item for her, but I don't think you need to get one.

Use common sense here. What does Kayle need? Attack Speed, Attack Damage, and Ability Power.