D) The sum of the measures of the arcs from A to C, one passing through B and the other passing through D, is 360 degrees. E) Angle CDA measures 80 degrees. F) Angle CDA measures 100 degrees.
D) The sum of the measures of the arcs from A to C, one passing through B and the other passing through D, is 360 degrees. E) Angle CDA measures 80 degrees. F) Angle CDA measures 100 degrees.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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The image shows a quadrilateral inscribed in a circle. Select all true statements.
A) The arc from B to D passing through C measures 70 degrees.
B) The arc from A to C passing through D measures 200 degrees.
C) Angle BCD measures 140 degrees.
D) The sum of the measures of the arcs from A to C, one passing through B and the other passing through D, is 360 degrees.
E) Angle CDA measures 80 degrees.
F) Angle CDA measures 100 degrees.
![### Inscribed Quadrilateral in a Circle
In the given diagram, a cyclic quadrilateral \(ABCD\) is inscribed in a circle with center \(O\). The vertices \(A\), \(B\), \(C\), and \(D\) lie on the circumference of the circle.
Key features of the diagram include:
- The circle's center is denoted by \(O\).
- The quadrilateral \(ABCD\) is cyclic, meaning all its vertices are points on the circle.
- The interior angles \( \angle DAB \) and \( \angle DCB \) are marked as \(70^\circ\) and \(100^\circ\) respectively.
### Important Properties:
1. **Opposite Angles of a Cyclic Quadrilateral**:
- In a cyclic quadrilateral, the sum of the opposite angles is always \(180^\circ\). Therefore, \( \angle DAB + \angle DCB = 180^\circ \).
### Conclusion:
Given the angles in the quadrilateral:
\[ \angle DAB = 70^\circ \]
\[ \angle DCB = 100^\circ \]
By the property of opposite angles in a cyclic quadrilateral:
\[ 70^\circ + 110^\circ = 180^\circ \ ( \because \ \angle DAB \ \& \ \angle DCB) \]
Similarly:
\[ 100^\circ + 80^\circ = 180^\circ \ ( \because \ \angle ABC \ \& \ \angle CDA) \]
This confirms the diagram is mathematically consistent for a cyclic quadrilateral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5400a9d8-f651-4eb8-82bb-99e479bab37f%2Fb7c734e6-520f-45f2-957b-b9dcddb056aa%2Frksnft3_processed.png&w=3840&q=75)
Transcribed Image Text:### Inscribed Quadrilateral in a Circle
In the given diagram, a cyclic quadrilateral \(ABCD\) is inscribed in a circle with center \(O\). The vertices \(A\), \(B\), \(C\), and \(D\) lie on the circumference of the circle.
Key features of the diagram include:
- The circle's center is denoted by \(O\).
- The quadrilateral \(ABCD\) is cyclic, meaning all its vertices are points on the circle.
- The interior angles \( \angle DAB \) and \( \angle DCB \) are marked as \(70^\circ\) and \(100^\circ\) respectively.
### Important Properties:
1. **Opposite Angles of a Cyclic Quadrilateral**:
- In a cyclic quadrilateral, the sum of the opposite angles is always \(180^\circ\). Therefore, \( \angle DAB + \angle DCB = 180^\circ \).
### Conclusion:
Given the angles in the quadrilateral:
\[ \angle DAB = 70^\circ \]
\[ \angle DCB = 100^\circ \]
By the property of opposite angles in a cyclic quadrilateral:
\[ 70^\circ + 110^\circ = 180^\circ \ ( \because \ \angle DAB \ \& \ \angle DCB) \]
Similarly:
\[ 100^\circ + 80^\circ = 180^\circ \ ( \because \ \angle ABC \ \& \ \angle CDA) \]
This confirms the diagram is mathematically consistent for a cyclic quadrilateral.
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