To solve the problem of finding the measures of two complementary angles where one angle is 15 degrees less than twice the measure of the other, we need to set up a system of equations based on the given conditions.
1. Complementary Angles Condition:
The sum of two complementary angles is [tex]\(90^\circ\)[/tex]. Hence, we can write the first equation as:
[tex]\[
a + b = 90
\][/tex]
Here, [tex]\(a\)[/tex] and [tex]\(b\)[/tex] represent the measures of the first and second angles, respectively.
2. Given Relationship Between Angles:
The measure of the first angle is 15 degrees less than twice the measure of the second angle. We can express this relationship as:
[tex]\[
a = 2b - 15
\][/tex]
Therefore, the system of equations that satisfies the given conditions is:
[tex]\[
\begin{cases}
a + b = 90 \\
a = 2b - 15
\end{cases}
\][/tex]
Now, looking at the provided options:
A.
[tex]\[
\begin{cases}
a + b = 90 \\
2b - 15 = a
\end{cases}
\][/tex]
This matches our set of equations.
B.
[tex]\[
\begin{cases}
a + b = 90 \\
a - 2b = -15
\end{cases}
\][/tex]
This does not match our set of equations.
C.
[tex]\[
\begin{cases}
a + b = 90 \\
2a - 15 = b
\end{cases}
\][/tex]
This does not match our set of equations.
D.
[tex]\[
\begin{cases}
a + b = 90 \\
2b + 15 = a
\end{cases}
\][/tex]
This does not match our set of equations.
Thus, the correct answer is:
A.
[tex]\[
a + b = 90
\][/tex]
[tex]\[
2b - 15 = a
\][/tex]
