# In a triangle, the angles are u°, v° and w°.?

In a triangle, the angles are u°, v° and w°. Determine the angles of the triangle if v is half the size of w and if u is the mean of v and w.

My book say that the answer are:

u=60°

v=40°

w=80°

But I dont know how to solve the problem, can someone help me?

Relevance

If v is half of w:

v = w/2

and u is the mean of v and w:

u = (v + w) / 2

And you know it's a triangle, the sum must be 180:

u + v + w = 180

We now have a system of three equations and three unknowns that we can solve.

Let's first substitute the expression for u in terms of v and w into from the second equation into the third:

(v + w) / 2 + v + w = 180

Let's multiply both sides by 2 to simplify:

v + w + 2v + 2w = 360

3v + 3w = 360

Divide both sides by 3:

v + w = 120

The first equation is v in terms of w, so substitute that into this equation and solve for w:

w/2 + w = 120

Again, multiply both sides by 2:

w + 2w = 240

3w = 240

w = 80

Now that we have w we can solve for v and u:

v + w = 120

v + 80 = 120

v = 40

u = (v + w) / 2

u = (40 + 80) / 2

u = 120 / 2

u = 60

u = 60, v = 40, w = 80

• In a triangle, the angles are u°, v° and w°.

Determine the angles of the triangle,

if v is half the size of w and if u is the mean of v and w.

u + v + w = 180°

(v + w)/2 + w/2 + w = 180°

v + 4w = 360°

3w - u = 180° = u + v + w

2w - 2u - v = 0

Solution:

u = 60°, v = 40°, w = 80°

• Call the angles 2x, 4x, (2x + 4x)/2 = 3x

2x + 4x + 3x = 9x = 180°, x = 20.

The angles are: 40°, 80°, 60°.