The PAT was implemented to help reduce the number of ties. It was assumed that XPs would be missed at times but of course now it is almost automatic. The two point conversion was added to make games more exciting by giving teams a chance to tie or win a game they would otherwise not win with an XP kick. Under NFL rules, a team does not have to attempt points after touchdown.
It is a scoring play used as a substitute for a point-after kick - the ball is advanced into the endzone like a touchdown play but it comes after the touchdown.
That is the correct spelling of "touchdown" (a football scoring play).
After scoring a touchdown the football team's coach opted to go for the 2-pt conversion instead of the traditional extra point try.
by throwing a ball and scoring a touchdown
They don't, the two point conversion is simply named a two point conversion where after a team scores a touchdown they attempt to get into the endzone once again, a safety is when the offense has possession of the ball and they are stopped and tackled in their own endzone
If a team attempts a two point conversion after scoring a touchdown and the ball is fumbled and recovered by the defense or if a pass is intercepted by the defense who then run the ball back to the other end zone the defense team will be credited with the two points. This is called a defensive conversion. Note that this rule only applies in college football. In the NFL, the defense can't score on a point-after attempt.
You can attempt a field goal after scoring a touchdown, or on any fourth down play within you range
point after touchdown
Scoring a field goal in American football cannot be done in conjunction with a touchdown. They are two independent ways of scoring. You may have meant can you score an "extra point" without a touchdown. In American football, extra points (1 or 2 depending on type of play), can only happen in conjunction with a touchdown. In Canadian football, they may be scored under circumstances not related to a touchdown.
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The scoring team is given 6 points for the touchdown. They also get the choice to kick an extra point or go for a two-point conversion after the touchdown takes place.
A touchdown scores six points. The "extra point" kick tacks on a seventh point, or the scoring team can attempt a "two-point conversion" for a total of 8 points.