Answer:
Yes since the triangles are congruent by ASA. See proofs and explanation below.
Explanation:
The cardio and free weights will have the same amount of space because triangles ABE and BCE are congruent by ASA or Angle-Side-Angle. We know that angle AEB and angle DEC are congruent since they are vertical angles. This is the first angle is ASA. We also know that the two triangles must be congruent because line segment DE is congruent to line segment EB since E is the midpoint of DB. This is the side in ASA. Lastly, the other angle of ASA can be proven because AB and DC are parallel lines. SInce they are parallel they have special angle relationships when the transversal DB intersects them. Since angle ABE and angle EDC are alternate interior angles, they are congruent. This means the triangles must be congruent. When triangles are congruent, then each triangle is the same shape and size. This means their area is the same too. Therefore, cardio and free weights have the same amount of space.
To write a two columns proof, start with given information, add new information, and then conclude.
Statement Reason
1. AB ║CD Given
2. E midpoint of DB Given
3. DE ≅ EB Definition of Midpoint
4. ∠ABE≅∠EDC Alternate Interior Angles
5. ∠AEB ≅ ∠DEC Vertical Angle Theorem
6. ΔABE≅ΔCDE ASA
Use the angle relationships above and listed in number 4 to solve for the angle measures.