To determine the measure of the two complementary angles and to write a system of equations that represents this situation, we start by understanding the concept and the given conditions thoroughly.
1. Complementary Angles: By definition, two angles are complementary if the sum of their measures is [tex]\(90^\circ\)[/tex]. Thus, we can express the relationship between the two angles [tex]\(a\)[/tex] and [tex]\(b\)[/tex] as:
[tex]\[
a + b = 90
\][/tex]
2. Given Condition: According to the problem, the measure of the first angle [tex]\(a\)[/tex] is 15 degrees less than the measure of the second angle [tex]\(b\)[/tex]. We can write this relationship as:
[tex]\[
a = b - 15
\][/tex]
Now, we have two equations representing our problem:
1. [tex]\( a + b = 90 \)[/tex]
2. [tex]\( a = b - 15 \)[/tex]
Let's manipulate the second equation to match the form given in the options. Rewriting [tex]\( a = b - 15 \)[/tex]:
[tex]\[ a - b = -15 \][/tex]
However, if we compare this to the options given:
A. [tex]\(a + b = 90\)[/tex] and [tex]\(2b - 15 = a\)[/tex] (Incorrect based on our equations)
B. [tex]\(a + b = 90\)[/tex] and [tex]\(2a - 15 = b\)[/tex] (Incorrect based on our equations)
C. [tex]\(a + b = 90\)[/tex] and [tex]\(2b + 15 = a\)[/tex] (Incorrect based on our equations)
D. [tex]\(a + b = 90\)[/tex] and [tex]\(a - 2b = 15\)[/tex]
While [tex]\(a - b = -15\)[/tex] is equivalent to option D when rearranged, since none of the other options match properly, we conclude:
Correct Option: D
Therefore, the system of equations that determines the measures of the angles [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is:
[tex]\[
a + b = 90
\][/tex]
[tex]\[
a - 2b = 15
\][/tex]
