Question

problem 20. (lesson 15) the difference between the measures of two complementary angles is 14. find the measure of both angles.

Explanation:

Step1: Define variables

Let the two complementary angles be \( x \) and \( y \), where \( x > y \). Since they are complementary, \( x + y = 90 \) (by the definition of complementary angles: their sum is \( 90^\circ \)). Also, the difference between them is 14, so \( x - y = 14 \).

Step2: Solve the system of equations

We have the system:
\[

$$\begin{cases} x + y = 90 \\ x - y = 14 \end{cases}$$

\]
Add the two equations together: \( (x + y)+(x - y)=90 + 14 \). Simplifying the left side: \( x + y+x - y = 2x \), and the right side: \( 104 \). So \( 2x=104 \), then \( x=\frac{104}{2}=52 \).

Step3: Find the other angle

Substitute \( x = 52 \) into \( x + y = 90 \): \( 52 + y = 90 \). Subtract 52 from both sides: \( y=90 - 52 = 38 \).

Answer:

52 and 38