Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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Find each value or measure
![### Geometry Problem: Angle Measurements in a Circle
**Figure:**
In the given diagram, a circle with center \( E \) has an inscribed quadrilateral \( ABCD \). The angles at points \( A \), \( B \), \( C \), and \( D \) are noted as follows:
- \(\angle A = 39^\circ\)
- \(\angle B = 61^\circ\)
- \(\angle D = 147^\circ\)
**Objective:**
Determine the measures of angles \( \angle A \), \( \angle B \), \( \angle C \), and \( \angle D \).
**Given Values:**
- \( m\angle A = \)
- \( m\angle B = \)
- \( m\angle C = \)
- \( m\angle D = \)
**Explanation:**
The circle contains a quadrilateral \( ABCD \) where three angles are provided. Verify knowledge of the theorem on cyclic quadrilaterals: **Opposite angles of a cyclic quadrilateral sum to \( 180^\circ \)**. Utilize this information to solve the problem.
**Solution Approach:**
1. Calculate \( \angle D \) directly since it is given.
2. Use the given angle measures and properties of cyclic quadrilaterals to find \( \angle C \).
3. Verify all angles and sums integrate correctly for a quadrilateral inscribed in a circle.
**Steps:**
1. **For given angle \( D \) (147°):**
\[
m\angle D = 147^\circ
\]
2. **From the theorem:**
\( \angle B + \angle D = 180^\circ \)
\[
m\angle C = 180^\circ - 147^\circ = 33^\circ
\]
3. **List of angles:**
\[
m\angle A = \)
\[
m\angle B = 61^\circ
\]
\[
m\angle C = 33^\circ
\]
\[
m\angle D = 147^\circ
\ ]
Complete all measures to assess alignment with the total sum of \( 360^\circ\) within the quadrilateral circle.
### Study Tips:
- Refresh on the properties of cyclic quadrilaterals for problem-solving.
- Practice problems with different given angles to ensure full comprehension.
- Use the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F903f2ee1-6c1b-4cd4-8e8b-4909c5aa82ce%2Ffed5b54c-cac3-491c-a7aa-8d82000be70d%2Fu6skcbm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem: Angle Measurements in a Circle
**Figure:**
In the given diagram, a circle with center \( E \) has an inscribed quadrilateral \( ABCD \). The angles at points \( A \), \( B \), \( C \), and \( D \) are noted as follows:
- \(\angle A = 39^\circ\)
- \(\angle B = 61^\circ\)
- \(\angle D = 147^\circ\)
**Objective:**
Determine the measures of angles \( \angle A \), \( \angle B \), \( \angle C \), and \( \angle D \).
**Given Values:**
- \( m\angle A = \)
- \( m\angle B = \)
- \( m\angle C = \)
- \( m\angle D = \)
**Explanation:**
The circle contains a quadrilateral \( ABCD \) where three angles are provided. Verify knowledge of the theorem on cyclic quadrilaterals: **Opposite angles of a cyclic quadrilateral sum to \( 180^\circ \)**. Utilize this information to solve the problem.
**Solution Approach:**
1. Calculate \( \angle D \) directly since it is given.
2. Use the given angle measures and properties of cyclic quadrilaterals to find \( \angle C \).
3. Verify all angles and sums integrate correctly for a quadrilateral inscribed in a circle.
**Steps:**
1. **For given angle \( D \) (147°):**
\[
m\angle D = 147^\circ
\]
2. **From the theorem:**
\( \angle B + \angle D = 180^\circ \)
\[
m\angle C = 180^\circ - 147^\circ = 33^\circ
\]
3. **List of angles:**
\[
m\angle A = \)
\[
m\angle B = 61^\circ
\]
\[
m\angle C = 33^\circ
\]
\[
m\angle D = 147^\circ
\ ]
Complete all measures to assess alignment with the total sum of \( 360^\circ\) within the quadrilateral circle.
### Study Tips:
- Refresh on the properties of cyclic quadrilaterals for problem-solving.
- Practice problems with different given angles to ensure full comprehension.
- Use the
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