Let
x = measure of angle TAP
y = measure of angle BRE
We're told that "angle BRE is its own complement" meaning that angle BRE added to itself is 90 degrees. In algebraic terms, we can say
y+y = 90
2y = 90
2y/2 = 90/2
y = 45
So angle BRE is 45 degrees. Note how 45+45 = 90.
We also know that angle TAP and angle BRE are supplementary. They add up to 180 degrees
x+y = 180
x+45 = 180 ... replace y with 45
x+45-45 = 180-45 ... subtract 45 from both sides
x = 135
Since we defined x to be the measure of angle TAP, this means that angle TAP is 135 degrees