Given a tennis court in the form of a rectangle ATEF and two people standing at points F and B such that [tex]\( EF = EB \)[/tex]:
1. Identifying the Shape and Properties:
- The tennis court is specified to be a rectangle ATEF. Rectangles have several key properties: four right angles (each measuring 90 degrees), and opposite sides are equal in length.
- [tex]\( EF = EB \)[/tex] also informs us that triangle EFB is isosceles with EF and EB as the equal sides.
2. Analyzing Triangle EFB:
- In an isosceles triangle, the angles opposite the equal sides are also equal. Therefore, the angles at points F and B, denoted as [tex]\( y \)[/tex], must be equal.
- Because the sum of all angles within any triangle is 180 degrees, and we know that EFB is an isosceles triangle, we can conclude the remaining angle (at point E) will also help in determining the measures of [tex]\( y \)[/tex].
3. Determining Angle [tex]\( y \)[/tex]:
- Since both angles at F and B are equal, and we know that the sum of angles [tex]\( y + y + \angle FEB \)[/tex] must equal 180 degrees, where [tex]\( \angle FEB \)[/tex] is the angle at point E within the triangle.
- Given [tex]\(\angle FEB = 90^\circ \)[/tex] (because it is part of the rectangle each internal angle sum being complementary to 90 degrees):
[tex]\[
y + y + 90^\circ = 180^\circ
\][/tex]
- This simplifies to:
[tex]\[
2y = 90^\circ
\][/tex]
[tex]\[
y = 45^\circ
\][/tex]
4. Determining Angle [tex]\( x \)[/tex]:
- Angle [tex]\( x \)[/tex] is the angle at point AEB. Since ATEF is a rectangle, measuring the sum of right angles at point E.
- By the property of angles in the rectangle, any internal angle at point E conforming the sum to forming a straight line (which is 90 degrees on both sides).
[tex]\[
\angle AEB = 90^\circ
\][/tex]
Therefore, the measures of angles [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are:
- [tex]\( x = 90^\circ \)[/tex]
- [tex]\( y = 45^\circ \)[/tex]
Finally, the measures are:
- [tex]\( x = 90^\circ \)[/tex]
- [tex]\( y = 45^\circ \)[/tex]
- [tex]\( y = 45^\circ \)[/tex]
Thus, the angles are [tex]\( (90^\circ, 45^\circ, 45^\circ) \)[/tex].
