If you press down on a toilet handle and it doesn't flush, the problem could be with the handle itself, or it could be with the linkage that connects the handle to other components inside the tank. A small bit of troubleshooting should uncover the cause, which could also involve the fill valve or the shutoff valve.
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If the handle is broken, you should immediately tell when you press it. The handle itself may be broken, which you can visually notice on the outside of the tank, which happens especially when the handle is plastic, and it somehow cracks. With continued use, the crack spreads even further until you press the handle and the crack is so wide that only a portion of the handle goes down, and the handle is unable to effectively start a flush. Replace a broken handle.
Broken Flush Lever
The handle is the outer component, but directly on the other side of the tank's front wall is a flush lever, which is connected to the handle. When you press the handle down, one end of the flush lever rises, helping start the flush. Often you can diagnose a broken flush lever by simply looking at the handle. If a handle that normally sits horizontally has now fallen into a vertical position, the lever is likely broken, although this handle position could also mean that you simply need to tighten the nut on the back of the handle. If the flush lever is broken, install a new flush lever with a new handle attached to it.
The chain that goes from the flush lever to the rubber flapper may have fallen off. Remove the tank and examine the end of the flush lever to determine if this has happened, and securely attach the chain back onto the hook on the flush lever or onto one of the flush lever holes. If you find that the chain is hooked onto the flush lever, ensure that the chain isn't too long, which would prevent the flapper from lifting properly. The chain should only have a minimum amount of slack. Cut off any excess chain links using a pair of snips.
The handle may seem like it isn't working properly to flush the toilet, but the cause could actually be from another source -- no water. If there is no tank water to flush down through the bowl, pressing the handle won't cause anything to happen. Check the shutoff valve, normally found under the left side of the toilet; it might be twisted all the way clockwise. Twist the valve counterclockwise, and see if water now fills the toilet tank. If the tank doesn't refill, check the fill valve for any debris clogging it. Turn the shutoff valve clockwise, remove the fill valve top, turn the shutoff valve counterclockwise and flush water through the fill valve for a few seconds. If the fill valve still doesn't work properly, replace it.