Solve the problem.
-A student is asked to construct a triangle that satisfies the following criteria. The sum of the first and second angles is 108° more than the measure of the third angle. If the measure of the third angle is subtracted from the
Measure of the second angle, the result is twice the measure of the first angle. Find the measure of each angle of
The triangle.
A) First angle is 42°, second angle is 104°, third angle is 34°
B) First angle is 40°, second angle is 110°, third angle is 30°
C) First angle is 36°, second angle is 108°, third angle is 36°
D) First angle is 34°, second angle is 108°, third angle is 34°

Question

Quizplus · Accepted Answer

Let the first angle be x, the second angle be y, and the third angle be z. 
From the first condition, we have:

x + y = z + 108

From the second condition, we have:

y - z = 2x

We know that the sum of angles in a triangle is 180°, so we can write:

x + y + z = 180

Substituting the first and second equations into this equation, we get:

z + 108 + 2z + z = 180
4z = 72
z = 18

Substituting z = 18 into the first equation, we get:

x + y = 126

Substituting z = 18 into the second equation, we get:

y - 18 = 2x

Now we have two equations with two variables. Solving for x and y, we get:

x = 36, y = 108

Therefore, the measure of each angle of the triangle is: 
First angle = 36°
Second angle = 108°
Third angle = 36°

The only choice with these angles is C. Let x be the measure of the first angle, y be the measure of the second angle, and z be the measure of the third angle. 
From the problem, we have: 
x + y = z + 108  (equation 1)
y - z = 2x     (equation 2)
Also, we know that the sum of the angles in a triangle is 180 degrees. Thus: 
x + y + z = 180  (equation 3) 
Now we can solve for x, y, and z. 
From equation 1: 
y = z + 108 - x 
Substitute into equation 2: 
(z + 108 - x) - z = 2x 
Simplify: 
108 = 3x + z 
Substitute into equation 3: 
x + (z + 108 - x) + z = 180 
Simplify: 
2z + 108 = 180 
z = 36 
Substitute back into equation 1: 
x + y = 144 
Substitute z = 36 into equation 2: 
y - 36 = 2x 
y = 2x + 36 
Substitute y = 2x + 36 into x + y = 144: 
x + (2x + 36) = 144 
3x = 108 
x = 36 
Now we have: 
x = 36 
y = 2x + 36 = 108 
z = 36 
Thus, the measures of the angles are 36°, 108°, and 36° which matches choice C.

Solve the Problem. -A Student Is Asked to Construct a Triangle That Satisfies