Here are some ideas to relate golf to geometry:
(1) shapes: the ball approximates a sphere. The "dimples"on the ball are circular. The shaft of the club is a cylinder. The face of the club is often trapezoidal. The fairways are roughly rectangular, while the sand traps are sometimes circular. The cup that the ball falls into is cylindrical.
(2) Angles: the face of the club is angled from the vertical (clubs for shorter distances have a larger angle than clubs like drivers for longer distances), a golfer should be aware of the proper angle to hit the ball for maximum flight; hitting the ball with the wrong rotation will cause the ball to fly at an angle to the desired direction (e.g. a "slice"); a golfer takes into account the speed and direction of the wind in order to hit the ball at the proper angle (e.g. vector addition; vectors are used to describe transformations especially translations.)
(3) Curves: the ball follows an approximately parabolic curve while in flight (where a parabola is a conic section or a section of a cone), the shaft of the club could represent a line segment as could the pole for the flag; if the surface of the green is not flat the ball will follow a contour or curved path.
If you also consider mini golf, you use reflections when hitting the ball off of the sides (frequently you can visualize multiple reflections from one hit.)
The golfers swing takes the head of the club through a circular arc ( a portion of the circumference of a circle.)