SOLUTION: In right triangle DEF, the measure of angle D is 90 degrees and the measure of angle F is 12 degrees less than twice the measure of angle E. Find the measure of angle E.

Hi, there--

THE PROBLEM:
In right triangle DEF, the measure of angle D is 90 degrees and the measure of angle F is 12 
degrees less than twice the measure of angle E. Find the measure of angle E.

A SOLUTION:
Let D be the measure of angle D
Let E be the measure of angle E
Let F be the measure of angle F

Let's write some equations that model the relationships between the angle measures.

"Angle D is 90 degrees" translates to the equation, D=90

"The measure of angle F is 12 degrees less than twice the measure of angle E" translates to
F = 2E - 12


To solve this problem, we use the mathematical fact that the sum of the measures of the 
interior angles of any triangle is 180 degrees, or
D + E + F = 180

We would like to have an equation with only one variable. We can make several substitutions.

Substitute 90 for D in the interior angle equation. Simplify.
D + E + F = 180
(90) + E + F = 180
E + F = 180 - 90
E + F = 90

Substitute 2E - 12 for F in the interior angle equation. Solve for E.
E + (2E - 12) = 90
E + 2E - 12 = 90
3E - 12 = 90
3E = 102
E = 34

The measure of angle E is 34 degrees. 

We use the interior angle equation to find the measure of angle F.
D + E + F = 180

Substitute 90 for D and 34 for E. Simplify.
90 + 34 + F = 180
124 + F = 180
F = 180 - 124
F = 56

The measure of angle F is 56 degrees.

Now we check our numbers against the original words of the problem. 

The measure of angle F is 12 less than twice the measure of angle E.
56 is 12 less than twice 34.
56 is 12 less than 68.
68-12=56 
CHECK!

Hope this helps!

Mrs. Figgy
math.in.the.vortex@gmail.com