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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=

°

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Answer 1

In a rhombus, the properties we can use include:

1. Opposite angles are equal.
2. Adjacent angles are supplementary (add up to 180°).
3. The diagonals bisect each other at right angles.

Given that \( m\angle DAB = 124° \):

1. To find \( m\angle ABC \):
- Since \( m\angle DAB + m\angle ABC = 180° \):
\[
m\angle ABC = 180° - m\angle DAB = 180° - 124° = 56°
\]

2. To find \( m\angle AEB \):
- The diagonal \( AE \) bisects the angles \( DAB \) and \( ABC \). Hence, \( m\angle AEB = \frac{1}{2}(m\angle ABC + m\angle DAB) \):
\[
m\angle AEB = \frac{1}{2}(56° + 124°) = \frac{1}{2}(180°) = 90°
\]

3. To find \( m\angle ADE \):
- The angle \( ADE \) is equal to the angle \( ABC \) (since opposite angles in a rhombus are equal). Thus:
\[
m\angle ADE = m\angle ABC = 56°
\]

Finally, summarizing the measurements:

- \( m\angle ABC = 56° \)
- \( m\angle AEB = 90° \)
- \( m\angle ADE = 56° \)