Geometry Problems Involving the Angles of a Triangle



First, let’s remember some triangle trivia:

An isosceles triangle has two equal sides and two equal angles.

A right triangle has one 90o angle.

If a triangle is neither isosceles nor right, we will call it a generic triangle.

A piece of trivia that is true for all triangles:

The sum of the three angles of any triangle is equal to 180 degrees.
 
 

Now let’s try a problem.
 
 

The largest angle of a triangle is 5 times as big as the smallest angle. The third angle is 12 degrees more than the smallest angle. Find the measure of all three angles.

  1. What are we asked to find in this problem? Look at the last sentence of the problem, and you will see that we want to find the measures of all three angles of the triangle.


    2.  
     

  2. To define the variables in this problem, we will start by drawing a picture of a generic triangle:

    Since the problem tells us a lot about the largest angle and the middle angle but not much about the smallest angle, we’ll call the smallest angle x degrees.

    The largest angle is 5 times the smallest angle, so the largest angle is 5x degrees.

    The middle angle is 12 degrees more than the smallest angle, so we’ll call the middle angle x +12 degrees. We’ll write all of this information in our picture:

  3. Our equation will come from the fact that the sum of the three angles of any triangle must be 180 degrees. This gives us:

    (5x) + (x + 12) + (x) = 180



  4. Now we’ll solve the equation:

    5x + x + 12 + x = 180

    7x + 12 = 180

    7x = 168

    x = 24

    So the smallest angle measures 24 degrees.


    5.  
     

  5. Have we finished the problem? Not quite. We still need to find the measures of the other two angles. We’ll refer back to our triangle to see that the largest angle measures 5x=5(24)=120o, and the middle angle measures x+12=24+12=36o.

    So the three angles measure 24o, 36o, and 120o.


    6.  
     

    Before we leave the problem, let’s check our answer with the words of the problem:
     

    words
    check
    The largest angle of a triangle is 5 times as big as the smallest angle. Is 120 five times as big as 24? Yes!
    The third angle is 12 degrees more than the smallest angle. Is 36 equal to 12 more than 24? Yes!
    This is a triangle: do the angles all add up to 180 degrees? 24 + 36 + 120 = 180 

    Our answer checks. We’re done!