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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 twice the measure of 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]$a - 2b = -15$[/tex]

C. [tex]$a + b = 90$[/tex]
[tex]$2a - 15 = b$[/tex]

D. [tex]$a + b = 90$[/tex]
[tex]$2b + 15 = a$[/tex]


Sagot :

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]