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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]
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]
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