Question
14 of 1914 of 19 Items choose the correct answers carefully and tell me why that is correct
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=
°
Skip to navigation
Answers
GPT-4o mini
In a rhombus, all sides are equal in length, and the opposite angles are equal. To solve the problem, let’s examine the known information and apply the properties of rhombuses:
1. **Given: \( m \angle DAB = 124^\circ \)**
### Step 1: Find \( m \angle ABC \)
Since ABCD is a rhombus, opposite angles are equal:
- \( m \angle DAB = m \angle BCD = 124^\circ \)
The consecutive angles in a rhombus are supplementary (they add up to \( 180^\circ \)):
- \( m \angle ABC + m \angle DAB = 180^\circ \)
- Substituting the known value:
\( m \angle ABC + 124^\circ = 180^\circ \)
- Solve for \( m \angle ABC \):
- \( m \angle ABC = 180^\circ - 124^\circ \)
- \( m \angle ABC = 56^\circ \)
### Step 2: Find \( m \angle AEB \)
To find \( m \angle AEB \), we need to consider the fact that diagonal \( AE \) in a rhombus bisects the angle \( DAB \):
- Since \( m \angle DAB = 124^\circ \), then:
- \( m \angle DAE = \frac{1}{2} m \angle DAB = \frac{1}{2} \times 124^\circ = 62^\circ \)
Now, \( m \angle AEB \) can be expressed as:
- \( m \angle AEB = m \angle DAE + m \angle ABC \)
- Substituting values:
\( m \angle AEB = 62^\circ + 56^\circ \)
- Therefore,
\( m \angle AEB = 118^\circ \)
### Step 3: Find \( m \angle ADE \)
Similarly, since diagonal \( AE \) bisects \( m \angle ABC \):
- \( m \angle ADE = \frac{1}{2} m \angle ABC = \frac{1}{2} \times 56^\circ = 28^\circ \)
### Summary of Answers:
- \( m \angle ABC = 56^\circ \)
- \( m \angle AEB = 118^\circ \)
- \( m \angle ADE = 28^\circ \)
This analysis relies on the properties of the rhombus regarding angle relationships and the fact that diagonals bisect the angles at the vertices.
1. **Given: \( m \angle DAB = 124^\circ \)**
### Step 1: Find \( m \angle ABC \)
Since ABCD is a rhombus, opposite angles are equal:
- \( m \angle DAB = m \angle BCD = 124^\circ \)
The consecutive angles in a rhombus are supplementary (they add up to \( 180^\circ \)):
- \( m \angle ABC + m \angle DAB = 180^\circ \)
- Substituting the known value:
\( m \angle ABC + 124^\circ = 180^\circ \)
- Solve for \( m \angle ABC \):
- \( m \angle ABC = 180^\circ - 124^\circ \)
- \( m \angle ABC = 56^\circ \)
### Step 2: Find \( m \angle AEB \)
To find \( m \angle AEB \), we need to consider the fact that diagonal \( AE \) in a rhombus bisects the angle \( DAB \):
- Since \( m \angle DAB = 124^\circ \), then:
- \( m \angle DAE = \frac{1}{2} m \angle DAB = \frac{1}{2} \times 124^\circ = 62^\circ \)
Now, \( m \angle AEB \) can be expressed as:
- \( m \angle AEB = m \angle DAE + m \angle ABC \)
- Substituting values:
\( m \angle AEB = 62^\circ + 56^\circ \)
- Therefore,
\( m \angle AEB = 118^\circ \)
### Step 3: Find \( m \angle ADE \)
Similarly, since diagonal \( AE \) bisects \( m \angle ABC \):
- \( m \angle ADE = \frac{1}{2} m \angle ABC = \frac{1}{2} \times 56^\circ = 28^\circ \)
### Summary of Answers:
- \( m \angle ABC = 56^\circ \)
- \( m \angle AEB = 118^\circ \)
- \( m \angle ADE = 28^\circ \)
This analysis relies on the properties of the rhombus regarding angle relationships and the fact that diagonals bisect the angles at the vertices.