Respuesta :
Answer:
1. [tex]m\angle KLP+m\angle PLM=180^{\circ}[/tex]
2. [tex]3x+m\angle PLM=180^{\circ}[/tex]
3. [tex]m\angle PLM=180^{\circ}-3x[/tex]
4. [tex]m\angle PMN=m\angle P+m\angle PLM[/tex]
5. [tex]2x+72^{\circ}=x+180^{\circ}-3x[/tex]
6. [tex]x=27^{\circ}[/tex]
Step-by-step explanation:
Please find the attached diagram for the complete question.
We are supposed to complete the given blanks.
1. [tex]m\angle KLP+m\angle PLM=...[/tex]
We can see from our given diagram that angle KLP and angle PLM are linear angles, so their measure will be equal to 180 degrees. Therefore, the correct expression for blank in 1st step would be [tex]180^{\circ}[/tex].
2. [tex]...+m\angle PLM=180^{\circ}[/tex]
Now, we will substitute the measure of angle angle KLP in our equation. We can see that measure of angle angle KLP is [tex]3x[/tex]. Therefore, the correct expression for blank in 2nd step would be [tex]3x[/tex].
3. [tex]m\angle PLM=180^{\circ}...[/tex]
Our next step is to find the measure of angle PLM in terms of x by subtracting [tex]3x[/tex] from both sides as:
[tex]3x-3x+m\angle PLM=180^{\circ}-3x[/tex]
[tex]m\angle PLM=180^{\circ}-3x[/tex]
Therefore, our 3rd step would be [tex]m\angle PLM=180^{\circ}-3x[/tex].
4. [tex]m\angle PMN=m\angle P+m\angle ...[/tex]
We can see that angle PMN is an exterior angle of our given triangle, so its measure will be equal to the sum of the opposite interior angles.
We can see that angle P and angle PLM are opposite interior angle of angle PMN, so we can set an equation as:
[tex]m\angle PMN=m\angle P+m\angle PLM[/tex]
Therefore, the correct expression for blank in 4th step would be [tex]]angle PLM[/tex].
5. [tex]...=...+180^{\circ}-3x[/tex]
Our next step is to substitute the values of angle PMN and angle P as given in the diagram.
[tex]2x+72^{\circ}=x+180^{\circ}-3x[/tex]
Therefore, our 5th step would be [tex]2x+72^{\circ}=x+180^{\circ}-3x[/tex].
6. [tex]x=[/tex]
Now, we need to solve for x using algebra as:
[tex]2x+72^{\circ}=x-3x+180^{\circ}[/tex]
[tex]2x+72^{\circ}=-2x+180^{\circ}[/tex]
[tex]2x+2x+72^{\circ}=-2x+2x+180^{\circ}[/tex]
[tex]4x+72^{\circ}=180^{\circ}[/tex]
[tex]4x+72^{\circ}-72^{\circ}=180^{\circ}-72^{\circ}[/tex]
[tex]4x=108^{\circ}[/tex]
[tex]\frac{4x}{4}=\frac{108^{\circ}}{4}[/tex]
[tex]x=27^{\circ}[/tex]
Therefore, the value of x is 27 degrees.
