The sum of two complementary angles is [tex]\( 90^\circ \)[/tex]. For one pair of complementary angles, the measure of the first angle is 15 less than the second angle. Write a system of equations that can be used to determine the measure of the first angle, [tex]\( a \)[/tex], and the measure of the second angle, [tex]\( b \)[/tex].

A. [tex]\( a + b = 90 \)[/tex]
[tex]\( 2b - 15 = a \)[/tex]

B. [tex]\( a + b = 90 \)[/tex]
[tex]\( 2a - 15 = b \)[/tex]

C. [tex]\( a + b = 90 \)[/tex]
[tex]\( 2b + 15 = a \)[/tex]

D. [tex]\( a + b = 90 \)[/tex]
[tex]\( a - 2b = 15 \)[/tex]



Answer :

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]