In a certain triangle the measure of one angle is triple the measure of a second angle but is 110 degrees less than the measure of the third angle. [The sum of the measures of three interior angles of a triangle is always 180 degrees.] What is the measure of each angle?
just write it algebraically:
b = 3a
b = c-110
a+b+c=180
a + 3a + (110+3a) = 180
7a = 70
a=10
b=30
c=140
To solve this problem, let's assign variables to the angles in the triangle. Let's call the first angle x, the second angle y, and the third angle z.
According to the problem, we're given three conditions:
1. "The measure of one angle is triple the measure of a second angle": This can be represented as x = 3y.
2. "The measure of one angle is 110 degrees less than the measure of the third angle": This can be represented as x = z - 110.
3. "The sum of the measures of three interior angles of a triangle is always 180 degrees": This can be represented as x + y + z = 180.
Now we can solve these equations to find the values of x, y, and z.
From condition 1, we can substitute the value of x in terms of y into equation 3:
3y + y + z = 180
4y + z = 180 ----(equation 1)
From condition 2, we can substitute the value of x in terms of z into equation 1:
z - 110 + y + z = 180
2z + y = 290 ----(equation 2)
Now we have a system of equations with two unknowns (y and z). We can solve these equations simultaneously to find the values of y and z.
Let's solve equations 1 and 2 using the method of substitution:
From equation 1:
4y + z = 180
z = 180 - 4y
Substitute this value of z into equation 2:
2(180 - 4y) + y = 290
360 - 8y + y = 290
-7y = -70
y = 10
Now that we have the value of y, substitute it back into equation 1 to find z:
4(10) + z = 180
40 + z = 180
z = 140
Finally, substitute the values of y and z back into equation 2 to find x:
2(140) + 10 = 290
280 + 10 = 290
x = 30
So, the measures of the angles in the triangle are:
Angle x = 30 degrees
Angle y = 10 degrees
Angle z = 140 degrees