the measure of three angles equal 180

Question

The measure of the smallest angle of a triangle is one-fifth the measure of the largest angle. The measure of the second angle is 62degrees° less than the measure of the largest angle. Find the measure of the angles of the triangle.how would i set this up??

Answer 1

You know that all of the the angles of the triangle (let's call them "x," "y," and "z" add up to 180 degrees.

 

So x+y+z = 180

 

All of the information that you are given about the other angles is relative to the "largest angle." So let's put the answers in terms of "x" where "x" is the "largest angle":  

 

 

The second angle (y) is 62 degrees less than the largest angle, so y = x-62. Now let's substitute that into the equation:

 

x + (x-62) + z = 180

 

The last angle (z) is one-fifth the measure of the largest angle (x), so z= (1/5)x. Now let's substitute that into the equation:

 

x + (x-62) + (1/5)x = 180

 

Now let's write it all out:

 

x + x - 62 + (1/5)x = 180

 

 

 

Time to simplify and solve for x. Start by adding 62 to both sides:

 

x + x -62 + (1/5)x = 180

         +62                +62

x + x + (1/5)x = 242

 

 

Now we need to get all of our x's together, so let's convert them all into fractions to make life easier:

 

1x = 5/5, so  x+x+(1/5)x is the same as:  (5/5)x + (5/5)x + (1/5)x =242

 

Since all of the x's are now in fifths, we can add them together:

 

(5+5+1)x = 242   so (11/5)x = 242

      5

 

Now we have to get x by itself, so divide both sides by (11/5). Note:  To divide by a fraction is the same as multiplying by its reciprocal:

 

(11/5)x = 242

(11/5)   = (11/5)

 

x = 110 degrees.

 

Now that you know angle x, you can solve for the others:

 

y = x - 62, so y = 110-62:  y = 48 degrees

 

z = (1/5)x, so z = (1/5)(110) = 22 degrees

 

Now check your answer:  x + y + z = 180?

 

110+48+22 =180

Answer 2

You need to set this up as simultaneous linear equations and substitute to solve.

 

call x, the smallest angle

y the next size

z the largest

 

Here's what you do know.

 

X+Y+Z= 180  ( that's true for all triangles)

X=.2 Z( 1/5) of the larger angle or Z

Y= Z-62

 

So, substitute the first equation above with the following:

 

.2Z + (z-62) + z=180

 

then combine to solve for Z

 

.2Z +z+Z= 180+62

 

2Z+.2Z=242

 

2.2Z=242

 

divide both sides by 2.2 to solve for Z

 

Z= 242/2.2 = 110

 

Now, go back and plug in 11 wherever you have Z

 

Since 

Y= Z-62

 

Y=110-62 or 48

 

and since 110+48+x=180

 

you can figure out that X= 180-158=22

 

so,

x=22  ( which is 1/5 of 110)

y=48

z=110