Question

MO bisects angle LMN , m angle LMO=8x-24 , and m angle NMO=3x+31 Solve for x and find m angle LMN . The diagram is not to scale.

Answer 1

We will see the how we can manipulate interioir angles in terms of variables and measure their degree of angle mathematically.

We are given the following statements.

An angle LMN is bisected by a ray ( MO ). This statement can be read as that we have an unknown large angle < LMN which is divided into two halves ( bisected ) by a ray with common vertex at M originating from point O.

We can represent the above statement in a graphical form for better understanding as follows:

We see that the angle < LMN which unknown can be given a symbol as follows:

`\angle LMN\text{ = }\theta`

The ray MO divided the above angle in to two halves. The two created constituent angles are marked and named as follows:

`\angle LMO\text{ \& }\angle NMO`

Since the larger angle is halved then the two consituent angles must be equal with each other and:

`\angle LMO\text{ = }\angle NMO\text{ = }(\theta)/(2)`

Each of the created constituent angles are expressed in terms of variable ( x ) as follows:

`\begin{gathered} \angle LMO\text{ = 8x - 24} \\ \angle NMO\text{ = 3x + 31} \end{gathered}`

We can use the angle bisection relation that we wrote above and equate the two equally halved angled as follows:

`\begin{gathered} \angle LMO\text{ = }\angle NMO \\ 8x\text{ - 24 = 3x + 31} \end{gathered}`

Now we can solve the above equation for the variable ( x ) as follows:

`\begin{gathered} 5x\text{ = 55} \\ \textcolor{#FF7968}{x=}\text{\textcolor{#FF7968}{ 11 degrees}} \end{gathered}`

Using the value of ( x ) we can now evaluate the anlge measures of constituent angles as follows:

`\begin{gathered} \angle LMO\text{ = 8}\cdot(11)\text{ - 24 = 88 - 24 = 64 degrees} \\ \angle NMO\text{ = 3}\cdot(11)\text{ + 31 }=\text{ 33 + 11 = 64 degrees} \end{gathered}`

Now again we can use the anlge bisection relationship to determine the measure of the larger angle LMN as follows:

`\begin{gathered} \angle LMN\text{ = 2}\cdot\angle LMO\text{ = 2}\cdot\angle NMO \\ \angle LMN\text{ = 2}\cdot64 \\ \textcolor{#FF7968}{\angle LMN}\text{\textcolor{#FF7968}{ = }}\textcolor{#FF7968}{128}\text{\textcolor{#FF7968}{ degrees}} \end{gathered}`

Hence, the answers are:

`\begin{gathered} \textcolor{#FF7968}{x}\text{\textcolor{#FF7968}{ = 11 degrees}} \\ \textcolor{#FF7968}{m\angle}\text{\textcolor{#FF7968}{LMN = 128 degrees}} \end{gathered}`