Sure, let's break down the solution to each part of the question step-by-step.
### Problem 1: Complementary Angles
We are given two angles, [tex]\(2p^{\circ}\)[/tex] and [tex]\((p + 15)^{\circ}\)[/tex], that are complementary.
Step 1: Understand that complementary angles sum to 90 degrees.
[tex]\[ 2p + (p + 15) = 90 \][/tex]
Step 2: Combine like terms.
[tex]\[ 2p + p + 15 = 90 \][/tex]
[tex]\[ 3p + 15 = 90 \][/tex]
Step 3: Isolate the term with the variable by subtracting 15 from both sides.
[tex]\[ 3p = 75 \][/tex]
Step 4: Solve for [tex]\(p\)[/tex] by dividing by 3.
[tex]\[ p = 25 \][/tex]
Step 5: Find the measures of the two angles by substituting [tex]\(p\)[/tex] back into the expressions for the angles.
[tex]\[ \text{First angle} = 2p = 2 \cdot 25 = 50^{\circ} \][/tex]
[tex]\[ \text{Second angle} = p + 15 = 25 + 15 = 40^{\circ} \][/tex]
So, the two complementary angles are 50 degrees and 40 degrees.
### Problem 2: Supplementary Angles
We are given two angles, [tex]\(x^{\circ}\)[/tex] and [tex]\(\frac{x^{\circ}}{4}\)[/tex], that are supplementary.
Step 1: Understand that supplementary angles sum to 180 degrees.
[tex]\[ x + \frac{x}{4} = 180 \][/tex]
Step 2: Combine like terms by rewriting the fraction with a common denominator.
[tex]\[ \frac{4x}{4} + \frac{x}{4} = 180 \][/tex]
[tex]\[ \frac{5x}{4} = 180 \][/tex]
Step 3: Clear the fraction by multiplying both sides by 4.
[tex]\[ 5x = 720 \][/tex]
Step 4: Solve for [tex]\(x\)[/tex] by dividing by 5.
[tex]\[ x = 144 \][/tex]
Step 5: Find the measures of the two angles by substituting [tex]\(x\)[/tex] back into the expressions for the angles.
[tex]\[ \text{First angle} = x = 144^{\circ} \][/tex]
[tex]\[ \text{Second angle} = \frac{x}{4} = \frac{144}{4} = 36^{\circ} \][/tex]
So, the two supplementary angles are 144 degrees and 36 degrees.
Therefore, the answers are:
- Complementary Angles: 50 degrees and 40 degrees.
- Supplementary Angles: 144 degrees and 36 degrees.
