Question

In the diagram above, quadrilateral ABCD is a rhombus. If the m ∠DAB
= 124 °
find the measurements below.

(3 points)
m∠ABC=

°

m∠AEB=

°

m∠ADE

Answer 1

In a rhombus, opposite angles are equal, and adjacent angles are supplementary. This means that:

1. **For the angles:**
- Since \( m\angle DAB = 124° \), we have:
\[
m\angle ABC = m\angle DAB = 124°
\]
- Thus, \( m\angle ABC = 124° \).

2. **Calculating the other angles:**
- The angles adjacent to \( m\angle DAB \) and \( m\angle ABC \) are supplementary:
\[
m\angle ABC + m\angle BCD = 180°
\]
- Therefore:
\[
m\angle BCD = 180° - 124° = 56°
\]
- And since opposite angles in a rhombus are equal:
\[
m\angle ABC = m\angle DAB = 124° \quad \text{and} \quad m\angle BCD = m\angle DAB = 56°
\]

3. **Finding angle \( m\angle AEB \)**:
- The points A, B, D, and E give a diagonal split through vertex angles. The angle \( m\angle AEB \) sits within the triangle ABE formed by the diagonals intersecting at point E.
- In a rhombus, the diagonals bisect the angles:
\[
m\angle AEB = \frac{m\angle DAB}{2} = \frac{124°}{2} = 62°
\]

4. **Finding angle \( m\angle ADE \)**:
- Using the same property of bisected angles, angle \( m\angle ADE \) is half of angle DAB:
\[
m\angle ADE = \frac{m\angle DAB}{2} = \frac{124°}{2} = 62°
\]

Thus, we can summarize the answers as follows:
- \( m\angle ABC = 56° \)
- \( m\angle AEB = 62° \)
- \( m\angle ADE = 62° \)