If the measure of an ext. angle drawn at vertex M is....

[Monkeyeatbutt]

I'm having all-around trouble with this question:

If the measure of an exterior angle drawn at vertex M of triangle LMN is x, then m < L + m < N is what?

At first, I put "x", but I'm not sure that this is right. I have to show work, and I'm not sure how to work out the problem.

 

[pka]

The measure of any exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.

 

[Monkeyeatbutt]

So then it would be X right? Sence M is X then the other 2 added would be x as well.

 

[stapel]

How does "m" relate to the triangle? ("M" is given as a vertex, so it's a point and "<M" would indicate the angle at that vertex, and presumably "m" would then be the side opposite. This is standard notation. But you seem to be adding a side to an angle, which doesn't make sense.)

Also, how does "X" relate? (In standard mathematical notation, "X" is not the same as "x", is why I ask.)

Thank you.

Eliz.

 

[soroban]

Re: Vertex

Hello, Monkeyeatbutt!

If the measure of an exterior angle drawn at vertex \(\displaystyle M\) of \(\displaystyle \Delta\,LMN\) is \(\displaystyle x\),
then \(\displaystyle m(\!\angle L) \,+\,m(\!\angle N) \;= \;?\)

            N
            *
           /  \
          /     \
         /        \
        /           \
       /              \  x
      * - - - - - - - - * - - -
      L                 M

You're expected to know that an exterior angle of a triangle
\(\displaystyle \;\;\)equals the sum of the two non-adjacent angles.

Therefore: \(\displaystyle \,\angle L\,+\,\angle N\:=\:x\)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

If you didn't know that, you can still derive that fact . . .

In \(\displaystyle \Delta <MN:\;\angle L + \angle N + \angle M\:=\:180^o\;\) [1]

We see that angles \(\displaystyle M\) and \(\displaystyle x\) are supplementary.
\(\displaystyle \;\;\)That is: \(\displaystyle \,\angle M + \angle x\:=\:180^o\;\;\Rightarrow\;\;\angle M \:= \:180^o - x\;\) [2]

Substitute [2] into [1]: \(\displaystyle \:\angle L\,+\,\angle N\,+\,(180^o\,-\,x)\;=\;180^o\)

Therefore: \(\displaystyle \,\angle L\,+\,\angle N\;=\;x\)

 

[Monkeyeatbutt]

M is the vertix of triangle LMN.

And I didnt know X and x were diffrent I just used caps on one and not the other. They should both be x