Triangle KLM has a 120-degree angle at K and side KL is three-fourths as long as side LM. To the nearest tenth of a degree, find the sizes of the other two angles of KLM
Let x be the length of side LM.
Then side KL is (3/4)x.
Since we have a 120-degree angle at K, the sum of the other two angles is 180 - 120 = 60 degrees.
Let angle L be y and angle M be z.
Since the sum of the angles in a triangle is 180 degrees, we have
y + z = 60,
y + 120 + z = 180.
Substituting the first equation into the second equation, we get
60 + z = 180,
z = 120.
Since triangle KLM is an isosceles triangle (side KL = side LM), angle L is equal to angle M.
Thus, y = z = 120 degrees.
To find the size of the other two angles of KLM, we need to determine the measures of angles L and M.
Therefore, angle L and angle M are 120 degrees each. Answer: \boxed{120}.
To find the measures of the other two angles of triangle KLM, we can use the fact that the sum of the angles in a triangle is always 180 degrees.
Given:
Angle K = 120 degrees
Side KL = (3/4) * LM
Step 1: Find the measure of angle L.
Let the measure of angle L be x degrees.
Sum of angles in triangle KLM:
Angle K + Angle L + Angle M = 180 degrees
Replace the given values:
120 degrees + x + Angle M = 180 degrees
Step 2: Find the measure of angle M.
To find the measure of angle M, we need to find the length of side LM.
According to the given information, side KL is three-fourths as long as side LM.
Let the length of side LM be y.
Length of KL = (3/4) * LM
KL = (3/4) * y
Since side KL is three-fourths the length of side LM, we can write the equation:
KL = (3/4) * LM
(3/4)y = (3/4) * LM
Since KL = (3/4) * y and KL = LM, we can equate them:
(3/4)y = (3/4) * LM
Simplifying the equation:
y = LM
Therefore, side KL = side LM.
Now, we can replace the values into the equation from step 1:
120 degrees + x + Angle M = 180 degrees
Replace Angle M with 180 - (120 + x):
120 degrees + x + (180 - (120 + x)) = 180 degrees
Simplifying the equation:
120 degrees + x + 180 - 120 - x = 180 degrees
120 - 120 cancels out, and x - x cancels out, leaving:
x = 180 - 180 = 0 degrees
Therefore, angle L is 0 degrees.
Now, we can substitute the values of angles K, L, and M:
Angle K = 120 degrees,
Angle L = 0 degrees,
Angle M = 60 degrees.
So, the sizes of the other two angles in triangle KLM are 0 degrees and 60 degrees (to the nearest tenth of a degree).