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