To solve the problem, we need to use the properties of a rhombus and its angles.
- Properties of a Rhombus:
- All sides are equal in length.
- Opposite angles are equal.
- Adjacent angles are supplementary (add up to 180°).
Given that \( m∠DAB = 124° \):
-
Finding \( m∠ABC \):
- Since \( ABCD \) is a rhombus, \( m∠DAB \) and \( m∠ABC \) are opposite angles.
- Therefore, \( m∠ABC = m∠DAB = 124° \).
-
Finding \( m∠ADE \):
- To find the adjacent angle \( m∠ABC \), we use the property that adjacent angles are supplementary.
- \( m∠DAB + m∠ABC = 180° \)
- Thus, \( m∠ABC = 180° - m∠DAB = 180° - 124° = 56° \).
- Therefore, \( m∠ADE = m∠ABC = 56° \).
-
Finding \( m∠AEB \):
- Point E is the intersection of the diagonals of the rhombus. In a rhombus, the diagonals bisect the angles.
- Since \( m∠DAB = 124° \), \( m∠EAB = \frac{1}{2} m∠DAB = \frac{1}{2} (124°) = 62° \).
- Similarly, \( m∠ABE = \frac{1}{2} m∠ABC = \frac{1}{2} (56°) = 28° \).
- In triangle \( AEB \), we have: \[ m∠AEB = 180° - (m∠EAB + m∠ABE) = 180° - (62° + 28°) = 180° - 90° = 90°. \]
Final Measurements:
- \( m∠ABC = 56° \)
- \( m∠AEB = 90° \)
- \( m∠ADE = 124° \)
Summary of Answers
- \( m∠ABC = 56° \)
- \( m∠AEB = 90° \)
- \( m∠ADE = 124° \)
These calculations utilize the properties of a rhombus, specifically the relationships between its angles and the bisection by diagonals.