Quote:
In the diagram above, triangle WXY intersects rectangle JKLM at points D, E, Q, R, S, and T. If line WX is parallel to line JK, what is the measure of angle WYX?
This is the Data Sufficiency (DS) question where were are asked to learn the measure of an angle WYX. Let us analyze the data from each statement first and try to identify if the given data is enough:
Statement 1: (1) The sum of the measures of angles DQS and ERT is 260.
Let us divide the solution in several steps:
a) We know that angles DQS and ERT are 260 degrees.
Angle DQS is external angle for the angle WQD. Angle ERT is external angle to ERX. We know that \(internal angle = 180 degrees - external angle\)
In this case, angles \(WQD + ERX = (180 - DQS) + (180 - ERT) = 360 - DQS - ERT = 360 - (DQS + ERT)\)
As \(DQS + ERT = 260 degrees\), => \(WQD + ERX = 360 - 260 = 100 degrees\)
b) Also, one may note that angles WQD and SQL are vertical and vertical angles are equal. The same applies to the angles ERX and TRM.
In this case \(WQD + ERX = SQL + TRM = 100\)
c) Now, as triangles SQL and TRM are right triangles, we have 2 angles QLS and RMT each of 90 degrees. In this case, sum of angles QSL and RTM are equal to \((180 - 90 - SQL) + (180 - 90 - TRM) = 90 - SQL + 90 - TRM = 180 - (SQL +TRM) = 180 - 100 = 80\)
d) Angles QSL and YST are vertical and because of it they are equal. The same applies to angles RTM and STY.
Thus, \(QSL + RTM = YST + STY = 80\).
e) Since all angles of a triangle are equal to 180 degrees, angle \(SYT = 180 - 80 = 100\)
We found an angle SYT what was required by the task.
Please note that calculations in Data Sufficiency questions are sometimes unnecessary waste of time, and it is required to understand the concept without solving the task itself.
Sufficient.
Statement 2:(2) The sum of the measures of angles WQD and ERX is 100.
This is what we found at the end of step a during analysis of 1st statement. Thus statement 2 is enough.
Sufficient.
Both statements are sufficient.
Answer:
D