To find the measures of angles [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] given that [tex]\( X \)[/tex] is 3 times the measure of [tex]\( Y \)[/tex] and they are supplementary angles, we will follow these steps:
1. Understand the concept of supplementary angles: Supplementary angles are two angles whose measures add up to [tex]\( 180^\circ \)[/tex].
2. Set up the relationship: Let [tex]\( Y \)[/tex] be the measure of angle [tex]\( Y \)[/tex]. We are given that angle [tex]\( X \)[/tex] is 3 times the measure of angle [tex]\( Y \)[/tex]. Thus, we can write:
[tex]\[
X = 3Y
\][/tex]
3. Form the equation based on the supplementary relationship: Since [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are supplementary, their sum is [tex]\( 180^\circ \)[/tex].
[tex]\[
X + Y = 180^\circ
\][/tex]
4. Substitute [tex]\( X \)[/tex] in the equation: Replace [tex]\( X \)[/tex] with [tex]\( 3Y \)[/tex] in the supplementary angle equation:
[tex]\[
3Y + Y = 180^\circ
\][/tex]
5. Simplify and solve for [tex]\( Y \)[/tex]: Combine like terms and solve for [tex]\( Y \)[/tex]:
[tex]\[
4Y = 180^\circ
\][/tex]
[tex]\[
Y = \frac{180^\circ}{4}
\][/tex]
[tex]\[
Y = 45^\circ
\][/tex]
6. Calculate the measure of angle [tex]\( X \)[/tex]: Since [tex]\( X \)[/tex] is 3 times the measure of [tex]\( Y \)[/tex], we have:
[tex]\[
X = 3 \times 45^\circ
\][/tex]
[tex]\[
X = 135^\circ
\][/tex]
Thus, the measures of angles [tex]\( Y \)[/tex] and [tex]\( X \)[/tex] are [tex]\( 45^\circ \)[/tex] and [tex]\( 135^\circ \)[/tex], respectively.
