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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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Answers
GPT-4o mini
In a rhombus, the opposite angles are equal and the adjacent angles are supplementary (i.e., they add up to \(180^\circ\)).
Given that \( m \angle DAB = 124^\circ \):
1. **Finding \( m \angle ABC \)**:
\[
m \angle ABC = m \angle DAB = 124^\circ
\]
2. **Finding \( m \angle AEB \)**:
The angle \( AEB \) is formed between the diagonals. In a rhombus, the diagonals bisect each other at right angles. Therefore, \( \angle AEB \) is half the measure of angle \( ABC \):
\[
m \angle AEB = \frac{1}{2} \cdot m \angle ABC = \frac{1}{2} \cdot 124^\circ = 62^\circ
\]
3. **Finding \( m \angle ADE \)**:
Since \( m \angle DAB + m \angle ABC = 180^\circ \):
\[
m \angle ABC + m \angle DAB = 180^\circ
\]
Therefore,
\[
m \angle ABC = 180^\circ - 124^\circ = 56^\circ
\]
Thus,
\[
m \angle ADE = 56^\circ
\]
So, the final measurements are:
- \( m \angle ABC = 56^\circ \)
- \( m \angle AEB = 62^\circ \)
- \( m \angle ADE = 124^\circ \)
Given that \( m \angle DAB = 124^\circ \):
1. **Finding \( m \angle ABC \)**:
\[
m \angle ABC = m \angle DAB = 124^\circ
\]
2. **Finding \( m \angle AEB \)**:
The angle \( AEB \) is formed between the diagonals. In a rhombus, the diagonals bisect each other at right angles. Therefore, \( \angle AEB \) is half the measure of angle \( ABC \):
\[
m \angle AEB = \frac{1}{2} \cdot m \angle ABC = \frac{1}{2} \cdot 124^\circ = 62^\circ
\]
3. **Finding \( m \angle ADE \)**:
Since \( m \angle DAB + m \angle ABC = 180^\circ \):
\[
m \angle ABC + m \angle DAB = 180^\circ
\]
Therefore,
\[
m \angle ABC = 180^\circ - 124^\circ = 56^\circ
\]
Thus,
\[
m \angle ADE = 56^\circ
\]
So, the final measurements are:
- \( m \angle ABC = 56^\circ \)
- \( m \angle AEB = 62^\circ \)
- \( m \angle ADE = 124^\circ \)