Sure, let's find the pair of supplementary angles step by step.
1. Definition of Supplementary Angles: Supplementary angles are two angles whose sum adds up to 180 degrees.
2. Setup the equation:
Suppose we have two angles, [tex]\(x\)[/tex] degrees and [tex]\(\frac{x}{4}\)[/tex] degrees. According to the definition of supplementary angles:
[tex]\[ x + \frac{x}{4} = 180 \][/tex]
3. Combine Like Terms:
To solve for [tex]\(x\)[/tex], first combine the terms on the left-hand side:
[tex]\[ x + \frac{x}{4} = 180 \][/tex]
Express [tex]\(x\)[/tex] with a common denominator:
[tex]\[ \frac{4x}{4} + \frac{x}{4} = 180 \][/tex]
[tex]\[ \frac{4x + x}{4} = 180 \][/tex]
[tex]\[ \frac{5x}{4} = 180 \][/tex]
4. Isolate [tex]\(x\)[/tex]:
To find the value of [tex]\(x\)[/tex], multiply both sides of the equation by 4:
[tex]\[ 5x = 720 \][/tex]
Then divide by 5:
[tex]\[ x = \frac{720}{5} \][/tex]
[tex]\[ x = 144 \][/tex]
5. Find the Pair of Angles:
Now that we have [tex]\(x = 144\)[/tex], we can find the two angles. The first angle is:
[tex]\[ x = 144^\circ \][/tex]
The second angle is:
[tex]\[ \frac{x}{4} = \frac{144}{4} = 36^\circ \][/tex]
6. Validate the Solution:
Finally, let's check if their sum is indeed 180 degrees:
[tex]\[ 144^\circ + 36^\circ = 180^\circ \][/tex]
Therefore, the pair of supplementary angles are:
[tex]\[ 144^\circ \text{ and } 36^\circ \][/tex]
