Let's solve the problem step-by-step:
1. Understand the Geometry of the Problem:
- We have a rectangle [tex]\( ATEF \)[/tex].
- Points [tex]\( F \)[/tex] and [tex]\( B \)[/tex] are such that [tex]\( EF = EB \)[/tex].
2. Identify Key Properties:
- Since [tex]\( EF = EB \)[/tex], triangle [tex]\( EFB \)[/tex] is an isosceles triangle.
- In an isosceles triangle, the angles opposite the equal sides are also equal. Hence, [tex]\( \angle EFB \)[/tex] (angle [tex]\( x \)[/tex] at [tex]\( F \)[/tex]) and [tex]\( \angle EBF \)[/tex] (angle [tex]\( y \)[/tex] at [tex]\( B \)[/tex]) are equal.
- Let's denote these angles as [tex]\( a \)[/tex]; i.e., [tex]\( \angle EFB = a \)[/tex] and [tex]\( \angle EBF = a \)[/tex].
3. Sum of Angles in Triangle [tex]\( EFB \)[/tex]:
- We know that the sum of the internal angles of any triangle is always 180 degrees.
- In triangle [tex]\( EFB \)[/tex]:
[tex]\[
\angle E + \angle EFB + \angle EBF = 180^\circ
\][/tex]
Given:
[tex]\[
\angle E + a + a = 180^\circ
\][/tex]
Simplifying,
[tex]\[
\angle E + 2a = 180^\circ
\][/tex]
4. Determine [tex]\( \angle E \)[/tex]:
- Since [tex]\( ATEF \)[/tex] is a rectangle, all the internal angles in a rectangle are 90 degrees.
- Therefore, [tex]\( \angle E = 90^\circ \)[/tex].
5. Solve for [tex]\( a \)[/tex]:
- Substitute [tex]\( \angle E = 90^\circ \)[/tex] into the equation from step 3:
[tex]\[
90^\circ + 2a = 180^\circ
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
2a = 180^\circ - 90^\circ
\][/tex]
[tex]\[
2a = 90^\circ
\][/tex]
[tex]\[
a = 45^\circ
\][/tex]
6. Conclusion:
- The measures of angles [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are [tex]\( 45^\circ \)[/tex] each.
Thus, the final measures for the angles are [tex]\( \angle x = 45^\circ \)[/tex] and [tex]\( \angle y = 45^\circ \)[/tex].
