Let's solve the given problems step-by-step to find the values of the angles.
### Part 1: Finding the Angles in the Linear Pair
1. Understanding the Linear Pair:
Two angles form a linear pair if they are adjacent and their measures add up to 180 degrees. So, we have:
[tex]\[
2x + 3x = 180^\circ
\][/tex]
2. Solving for [tex]\( x \)[/tex]:
Combine the like terms on the left side of the equation:
[tex]\[
5x = 180^\circ
\][/tex]
Divide both sides by 5 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{180^\circ}{5} = 36^\circ
\][/tex]
3. Finding Each Angle:
Using the value of [tex]\( x \)[/tex]:
- The first angle is [tex]\( 2x \)[/tex]:
[tex]\[
2x = 2 \times 36^\circ = 72^\circ
\][/tex]
- The second angle is [tex]\( 3x \)[/tex]:
[tex]\[
3x = 3 \times 36^\circ = 108^\circ
\][/tex]
So, the angles in the linear pair are [tex]\( 72^\circ \)[/tex] and [tex]\( 108^\circ \)[/tex].
### Part 2: Finding the Complementary Angle
1. Understanding Complementary Angles:
Two angles are complementary if their measures add up to 90 degrees. We are given:
[tex]\[
y + 48^\circ = 90^\circ
\][/tex]
2. Solving for [tex]\( y \)[/tex]:
Subtract 48 degrees from both sides:
[tex]\[
y = 90^\circ - 48^\circ = 42^\circ
\][/tex]
So, the complementary angle [tex]\( y \)[/tex] is [tex]\( 42^\circ \)[/tex].
### Summary of Results
- The angles in the linear pair are [tex]\( 72^\circ \)[/tex] and [tex]\( 108^\circ \)[/tex].
- The complementary angle [tex]\( y \)[/tex] is [tex]\( 42^\circ \)[/tex].
These are the final answers for the given problems.
