What is the appropriate next step in management for a postmenopausal woman presenting with painless vaginal bleeding?

The appropriate next step in management for a postmenopausal woman presenting with painless vaginal bleeding is a transvaginal ultrasound. This imaging technique is non-invasive and is the first-line diagnostic tool used to evaluate the endometrial lining in postmenopausal women experiencing such symptoms. Given that postmenopausal bleeding can be a sign of serious conditions, including endometrial hyperplasia or cancer, a transvaginal ultrasound helps assess the thickness of the endometrium. If the ultrasound shows an abnormally thickened endometrial lining, further investigation may then be warranted, such as a hysteroscopy or endometrial biopsy.

Other options, such as hysteroscopy or colposcopy with endocervical curettage, are more invasive procedures and are typically reserved for cases where there is already clear evidence of pathology that warrants such steps. A repeat Pap smear is generally not indicated in this scenario because it is primarily used for screening cervical cancer rather than investigating the cause of postmenopausal bleeding. Thus, transvaginal ultrasound serves as the most appropriate and effective next step for initial evaluation in this context.