To solve the problem, we need to determine the measure of angle [tex]\( Y \)[/tex] given the conditions stated in the question. Let's proceed step-by-step:
1. Understand the Relationship:
- Let the measure of angle [tex]\( Y \)[/tex] be [tex]\( y \)[/tex] degrees.
- Angle [tex]\( X \)[/tex] is [tex]\( 80\% \)[/tex] of angle [tex]\( Y \)[/tex]. Therefore, the measure of angle [tex]\( X \)[/tex] is [tex]\( 0.8 \times y \)[/tex].
2. Complementary Angles:
- By definition, complementary angles add up to [tex]\( 90 \)[/tex] degrees.
- Since angles [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are complementary, we have:
[tex]\[
X + Y = 90^\circ
\][/tex]
3. Set Up the Equation:
- Substitute the expression for angle [tex]\( X \)[/tex] into the equation:
[tex]\[
0.8y + y = 90
\][/tex]
4. Combine Like Terms:
- Combine the terms involving [tex]\( y \)[/tex] on the left side:
[tex]\[
1.8y = 90
\][/tex]
5. Solve for [tex]\( y \)[/tex]:
- To isolate [tex]\( y \)[/tex], divide both sides of the equation by [tex]\( 1.8 \)[/tex]:
[tex]\[
y = \frac{90}{1.8}
\][/tex]
6. Calculate the Value:
- Perform the division:
[tex]\[
y = 50
\][/tex]
Based on our calculations, the measure of angle [tex]\( Y \)[/tex] is [tex]\( 50^\circ \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{50^\circ} \][/tex]
