Detecting a (statistically) significant change in time series trend Suppose I have two time series (Series1 and Series2) which are identical from timePeriod -200 to timePeriod 0. Say that in timePeriod 0 they are both equal to 100. Series1is equal to 200 at timePeriod 50. Series2 is equal to 250 at timePeriod 50. They both go up and down over time, but Series2 is always equal to Series1+timePeriod number. (Ex. at timePeriod 20, it equals Series1+20.)
Obviously there is a significant difference between the levels of these 2 series at timePeriod 50. However, is there a way I can test to see whether there is a (statistically) significant difference in the trends of these two series over time? (Preferably, the method could be used with different numbers and time periods)
 A: This is a useful text on "Abrupt change detection" by Michèle Basseville and Igor V. Nikiforov,.  I think it can answer the substance of your question.  I think the CUSUM methods might do well there.
EDIT:
I am not sure about the method you describe.  After some thought, it sounds like you are asking about a correspondence between the two.  If so then making a lag plot or a scatter plot can be useful.  
A: If you care the use of R, a selection of packages available are summarized in the CRAN task view on time series (https://cran.r-project.org/web/views/TimeSeries.html). Below is an excerpt:

Change point detection is provided in strucchange and strucchangeRcpp
(using linear regression models) and in trend (using nonparametric
tests). The changepoint package provides many popular changepoint
methods, and ecp does nonparametric changepoint detection for
univariate and multivariate series. changepoint.np implements the
nonparametric PELT algorithm, changepoint.mv detects changepoints in
multivariate time series, while changepoint.geo implements the
high-dimensional changepoint detection method GeomCP. Factor-augmented
VAR (FAVAR) models are estimated by a Bayesian method with FAVAR.
InspectChangepoint uses sparse projection to estimate changepoints in
high-dimensional time series. Rbeast provides Bayesian change-point
detection and time series decomposition. breakfast includes methods
for fast multiple change-point detection and estimation.

