Please help: Angles L and M are complementary, as are....

[Guest]

Angle L and angle M are complementary angles. Angle N and angle P are complementary angles. If the measurement of angle L = y - 2, the measurement of angle M = 2x + 3, the measurement of angle N = 2x - y, and the measurement of angle P = x - 1, find the values of x, y, measurement of angle L, measurement of angle M, measurement of angle N, and measurement of angle P.

 

[soroban]

Re: Please help: Angles L and M are complementary, as are...

Hello, Blair!

What part is giving you trouble?

\(\displaystyle \angle L\) and \(\displaystyle \angle M\) are complementary angles.
\(\displaystyle \angle N\) and \(\displaystyle \angle P\) are complementary angles.

If \(\displaystyle \angle L\:=\:y\,-\,2,\;\;\angle M\:=\:2x\,+\,3,\;\;\angle N\:=\:2x\,-\,y,\;\;\angle P\:=\:x\,-\,1\),

find: \(\displaystyle \,x,\;y,\;\angle L,\;\angle M,\;\angle N,\;\angle P.\)

\(\displaystyle \angle L\,+\,\angle M\:=\:90^o\;\;\Rightarrow\;\;(y\,-\,2)\,+\,(2x\,+\,3)\:=\:90\;\;\Rightarrow\;\;2x\,+\,y\:=\:89\;\) [1]

\(\displaystyle \angle N\,+\,\angle P\:=\:90^o\;\;\Rightarrow\;\;(2x\,-\,y)\,+\,(x\,-\,1)\:=\:90\;\;\Rightarrow\;\;3x\,-\,y\:=\:91\;\) [2]

Add [1] and [2]: \(\displaystyle \,5x \,=\,180\;\;\Rightarrow\;\;x\,=\,36\)

Substitute into [1]: \(\displaystyle \,2\cdot36\,+\,y\:=\:89\;\;\Rightarrow\;\;y\,=\,17\)

Then we have: .\(\displaystyle \begin{array}{cccc}\angle L\:=\:y\,-\,2\:=\:17\,-\,2\:=\:15 \\
\angle M\:=\:2x\,+\,3\:=\:2\cdot36\,+\,3\:=\:75 \\
\angle N\:=\:2x\,-\,y\:=\:2\cdot36\,-\,17\:=\:55 \\
\angle P\:=\:x\,-\,1\:=\:36\,-\,1\:=\:35\end{array}\)