There are 1,000 fish in a lake. each year the population declines by 15%, but afterward, the lake is restocked with 500 additional fish. write a recursive rule to find the number of fish in the lake for any year.

Problem 8

If angles 'a' and b are a linear pair, then a+b = 180

This is the linear pair theorem. It's not a trick question.

A linear pair of angles is any angles that are adjacent and supplementary. Supplementary angles, by definition, always add to 180.

Problem 9

Angles BAT and CAP are known as vertical angles.

Vertical angles form whenever you have two lines cross like this. Vertical angles are always opposite one another and they are always the same measure.

We use the aptly named vertical angles theorem which has a very short proof, but I'll skip over that.

Problem 10

Notice how in both equation, we're involving angle A

We have

A+B = 180

A+C = 180

Then those directly lead to B = C.

We can use elimination to get B-C = 180-180 which becomes B-C = 0, and solves to B = C. In this case, the "A"s cancel out when subtracting straight down.

We use the supplement theorem to wrap everything up, or to simplify things in a single sentence/phrase. The theorem says that if two angles are supplementary to a third fixed angle, then those two first angles must be the same angle.

In other words, if x and y are supplementary to z (x is supplementary to z and y is supplementary to z), then x = y must be the case.

Side note: We never use choice C at all for any of problems 8 through 10