DEF and ABC are complementary angles and ABC is nine times as large as DEF. Determine the measure of each angle.
DEF + ABC = 90
DEF + 9DEF = 90
10DEF = 90
DEF = 9
ABC = 81
The shorter leg of a 30°-60°-90° triangle is 6. What is the length of the other leg?
To solve this problem, let's start by assigning variables to the angles. Let's say DEF is represented by x degrees.
Since DEF and ABC are complementary angles, the sum of their measures will be 90 degrees.
Therefore, we can write the equation:
x + ABC = 90
We also know that ABC is nine times as large as DEF, so we can write another equation:
ABC = 9x
Now we have a system of two equations:
x + 9x = 90 (substituting ABC's value from the second equation into the first equation)
10x = 90
x = 9
Now we have found the measure of DEF, which is x = 9 degrees.
To find the measure of ABC, we can substitute this value back into the second equation:
ABC = 9(9) = 81 degrees.
Therefore, DEF measures 9 degrees, and ABC measures 81 degrees.