Answer:
The value of x is 15. The measure of angle L and K is 45.7 degree. The measure of angle M is 77.6 degree.
Step-by-step explanation:
It is given that
[tex]\angle L\cong \angle K[/tex]
Since two angles are congruent, therefore we can say that the triangle KLM is isosceles triangle.
The sides KM and LM are congruent.
[tex]3x+23=7x-37[/tex]
[tex]60=4x[/tex]
[tex]15=x[/tex]
The value of x is 15.
The length of side KL is
[tex]KL=9x-40=9(15)-40=95[/tex]
The length of side KM and LM is
[tex]KM=LM=7x-37=7(15)-37=68[/tex]
Therefore length of sides are 68, 68 and 95.
Use Law of cosine to find the measure of each angle.
[tex]a^2=b^2+c^2-2bc\cos A[/tex]
Apply this formula according to the angle.
[tex]L=K=\cos ^{-1}(\frac{(KM)^2+(KL)^2-(LM)^2}{2(KM)(KL)}) =\cos ^{-1}(\frac{68^2+95^2-68^2}{2(68)(95)})=45.69^{\circ}[/tex]
Therefore the measure of angle L and K is 45.7 degree.
The measure of M is calculated as,
[tex]M=\cos ^{-1}(\frac{(LM)^2+(KM)^2-(KL)^2}{2(LM)(KM)}) =\cos ^{-1}(\frac{68^2+68^2-95^2}{2(68)(68)})=77.62^{\circ}[/tex]
Therefore the measure of angle M is 77.6 degree.
Answer:
x=15, ∠L=∠K=45.7° and ∠M is 77.6°.
Step-by-step explanation:
Given information: ∠L≅∠K, KL = 9x - 40, LM = 7x - 37, & KM = 3x + 23.
Two angles are congruent, so triangle KLM is an isosceles triangle. Corresponding adjacent sides of congruent angles are equal.
The sides KM and LM are congruent.
[tex]3x+23=7x-37[/tex]
Isolate variable terms.
[tex]3x-7x=-23-37[/tex]
[tex]-4x=-60[/tex]
Divide both sides by -4.
[tex]x=15[/tex]
The value of x is 15.
The length of sides of triangle KLM are
[tex]KL=9x-40=9(15)-40=95[/tex]
[tex]KM=LM=3(15)+23=45+23=68[/tex]
Therefore length of sides are 68, 68 and 95.
Law of cosine
[tex]a^2=b^2+c^2-2bc\cos A[/tex]
Using the above formula we get
[tex]K=L=\cos^{-1}(\frac{KM^2+KL^2-LM^2}{2(KM)(KL)})=\cos^{-1}(\frac{68^2+95^2-68^2}{2(68)(95)})=45.69^{\circ}[/tex]
[tex]A=\cos^{-1}(\frac{LM^2+KM^2-KL^2}{2(LM)(KM)})=\cos^{-1}(\frac{68^2+68^2-95^2}{2(68)(68)})=77.62^{\circ}[/tex]
Therefore, ∠L=∠K=45.7° and ∠M is 77.6°.
