Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

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]


Sagot :

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]