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

To find the measurements of the angles in rhombus ABCD, we can use the properties of rhombuses.

1. In a rhombus, all sides are equal, and opposite angles are equal. The sum of the interior angles of any quadrilateral is 360°.

2. Given that \( m \angle DAB = 124° \), we know that:
- \( m \angle ABC = 124° \) (since opposite angles in a rhombus are equal)

3. To find \( m \angle BCD \) and \( m \angle CDA \):
- Since \( m \angle DAB + m \angle ABC + m \angle BCD + m \angle CDA = 360° \)
- We also know \( m \angle BCD = m \angle CDA \) (since they are opposite angles).

Let \( x = m \angle BCD = m \angle CDA \). Therefore:
\[
124° + 124° + x + x = 360°
\]
\[
248° + 2x = 360°
\]
\[
2x = 360° - 248° = 112°
\]
\[
x = 56°
\]
So, \( m \angle BCD = 56° \) and \( m \angle CDA = 56° \).

4. To find \( m \angle AEB \), since E is the intersection of the diagonals of rhombus ABCD, and the diagonals bisect each other at right angles, we can know that:
- Angles \( m \angle AEB = \frac{1}{2} m \angle DAB\) since diagonals bisect the angles.

Therefore,
\[
m \angle AEB = \frac{1}{2} \times 124° = 62°
\]

5. Finally, to find \( m \angle ADE \), we can know that \( m \angle ADE = m \angle CDA \) since they are vertically opposite angles.
Thus,
\[
m \angle ADE = 56°
\]

In summary:
- \( m \angle ABC = 124° \)
- \( m \angle AEB = 62° \)
- \( m \angle ADE = 56° \)