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Question Which number answers are correct and why
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
To solve the problem involving rhombus ABCD, we need to understand some properties of rhombuses and the angles formed within them.
1. Since ABCD is a rhombus, all sides are of equal length, and opposite angles are equal. Also, the sum of the angles in any quadrilateral is 360°.
Given that \( m∠DAB = 124° \):
- Since opposite angles in a rhombus are equal:
\[
m∠BCD = m∠DAB = 124°
\]
- Now, to find \( m∠ABC \) and \( m∠ADC \), we can use the fact that adjacent angles in a rhombus are supplementary (they add up to 180°):
\[
m∠ABC + m∠DAB = 180°
\]
Therefore,
\[
m∠ABC = 180° - m∠DAB = 180° - 124° = 56°
\]
So,
\[
m∠ABC = 56°
\]
- Similarly, since \( m∠ADC \) is also adjacent to \( m∠DAB \):
\[
m∠ADC = 56°
\]
2. Next, let's consider the angle \( m∠AEB \). The point E is typically labeled for angle bisectors, diagonals, or other intersections. If E is the intersection point of the diagonals AC and BD in a rhombus:
- The diagonals of a rhombus bisect each other at right angles (90°).
Given that angle \( DAB \) is one of the angles formed by the diagonal segments, we can determine \( m∠AEB \) based on how diagonals behave:
Since diagonals bisect angles in a rhombus:
\[
m∠AEB = \frac{m∠DAB}{2} = \frac{124°}{2} = 62°
\]
3. Finally, to find \( m∠ADE \):
- Since \( m∠ADE \) is an angle formed from the diagonal and also adjacent to \( m∠DAB \), we can consider:
- Angle \( ADB \) can be found as it is part of the triangle formed by \( A, D, B \):
\[
m∠ADE = \frac{m∠DAB}{2} = \frac{124°}{2} = 62°
\]
So:
- \( m∠ABC = 56° \)
- \( m∠AEB = 62° \)
- \( m∠ADE = 62° \)
Final answers:
- \( m∠ABC = 56° \)
- \( m∠AEB = 62° \)
- \( m∠ADE = 62° \)
1. Since ABCD is a rhombus, all sides are of equal length, and opposite angles are equal. Also, the sum of the angles in any quadrilateral is 360°.
Given that \( m∠DAB = 124° \):
- Since opposite angles in a rhombus are equal:
\[
m∠BCD = m∠DAB = 124°
\]
- Now, to find \( m∠ABC \) and \( m∠ADC \), we can use the fact that adjacent angles in a rhombus are supplementary (they add up to 180°):
\[
m∠ABC + m∠DAB = 180°
\]
Therefore,
\[
m∠ABC = 180° - m∠DAB = 180° - 124° = 56°
\]
So,
\[
m∠ABC = 56°
\]
- Similarly, since \( m∠ADC \) is also adjacent to \( m∠DAB \):
\[
m∠ADC = 56°
\]
2. Next, let's consider the angle \( m∠AEB \). The point E is typically labeled for angle bisectors, diagonals, or other intersections. If E is the intersection point of the diagonals AC and BD in a rhombus:
- The diagonals of a rhombus bisect each other at right angles (90°).
Given that angle \( DAB \) is one of the angles formed by the diagonal segments, we can determine \( m∠AEB \) based on how diagonals behave:
Since diagonals bisect angles in a rhombus:
\[
m∠AEB = \frac{m∠DAB}{2} = \frac{124°}{2} = 62°
\]
3. Finally, to find \( m∠ADE \):
- Since \( m∠ADE \) is an angle formed from the diagonal and also adjacent to \( m∠DAB \), we can consider:
- Angle \( ADB \) can be found as it is part of the triangle formed by \( A, D, B \):
\[
m∠ADE = \frac{m∠DAB}{2} = \frac{124°}{2} = 62°
\]
So:
- \( m∠ABC = 56° \)
- \( m∠AEB = 62° \)
- \( m∠ADE = 62° \)
Final answers:
- \( m∠ABC = 56° \)
- \( m∠AEB = 62° \)
- \( m∠ADE = 62° \)