the sum of the measures of two acute angles is 110 degree.the suplement of the smaller anngle is 15 times the complement of the larger angle.what are the measures of the two angle?
x=smaller
y=larger
x+y=110
(180-x) = 15(90-y)
x=30
y=80
Let's break down the information given step-by-step to solve this problem.
Step 1: Let's assign variables to the two angles.
Let's call one angle "A" and the other angle "B".
Step 2: Write down the information we know.
The sum of the measures of two acute angles is 110 degrees:
A + B = 110
Step 3: Determine the relationship between the angles.
The supplement of the smaller angle is 15 times the complement of the larger angle.
The supplement of an angle is 180 degrees minus the angle.
The complement of an angle is 90 degrees minus the angle.
So, the relationship can be expressed as:
180 - A = 15(90 - B)
Step 4: Simplify the equation.
180 - A = 1350 - 15B
Add 15B to both sides:
180 + 15B - A = 1350
Rearrange the terms:
15B - A = 1350 - 180
15B - A = 1170
Step 5: Solve the system of equations.
We now have two equations:
A + B = 110 (equation 1)
15B - A = 1170 (equation 2)
We can solve this system of equations using the substitution method.
From equation 1, we can express A in terms of B:
A = 110 - B
Substitute this value of A into equation 2:
15B - (110 - B) = 1170
Simplify:
15B - 110 + B = 1170
Combine like terms:
16B - 110 = 1170
Add 110 to both sides:
16B = 1280
Divide by 16:
B = 80
Now, substitute the value of B back into equation 1 to find A:
A + 80 = 110
Subtract 80 from both sides:
A = 30
Step 6: Find the values of the two angles.
Angle A = 30 degrees
Angle B = 80 degrees
The two angles are 30 degrees and 80 degrees.
To solve this problem, we need to set up a system of equations based on the given information.
Let's call the measure of the smaller acute angle x, and the measure of the larger acute angle y.
From the first statement, we know that the sum of the measures of the two acute angles is 110 degrees:
x + y = 110
From the second statement, we know that the supplement of the smaller angle (180 - x) is 15 times the complement of the larger angle (90 - y):
180 - x = 15(90 - y)
Now, we have a system of two equations:
x + y = 110
180 - x = 15(90 - y)
To solve this system, we can use the method of substitution.
First, let's rearrange the second equation:
180 - x = 15(90 - y)
180 - x = 1350 - 15y
Next, let's simplify:
-x + 15y = 1170
Now, we can isolate x in terms of y:
x = 15y - 1170
Now, substitute the expression for x in the first equation:
15y - 1170 + y = 110
16y = 1280
y = 80
Substitute the value of y back into x = 15y - 1170:
x = 15(80) - 1170
x = 1200 - 1170
x = 30
Therefore, the measures of the two angles are:
x = 30 degrees
y = 80 degrees