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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![### Finding the Measures of Angles in a Hexagon
**Problem Statement:**
Given the hexagon below, find the measures of angles 1 through 7.
[In the image provided, a hexagon is presented with various angle measurements labeled.]
**Measurements and Internal Angles:**
1. Angle 3: 57°
2. Angle 4: 118°
3. Angle 5: 64°
4. Angle 6: 164°
5. Angle 7: 88°
**Steps for Finding Unknown Angles:**
1. **Hexagon Angle Sum Property:**
- The sum of the interior angles of a hexagon is given by the formula:
\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]
where \( n \) is the number of sides. For a hexagon ( \( n = 6 \) ):
\[
(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]
2. **Using Given Angles to Find Missing Angles:**
- Known angles are given as:
\[
57°, 118°, 64°, 164°, 88°
\]
- Calculate the sum of these known angles:
\[
57° + 118° + 64° + 164° + 88° = 491°
\]
3. **Finding the Remaining Angles (1 and 2):**
- Let angles 1 and 2 be \( x \) and \( y \) respectively. The sum of these angles can be found as:
\[
x + y + 491° = 720°
\]
\[
x + y = 720° - 491° = 229°
\]
- Since the relationship between \( x \) and \( y \) is not further defined in the problem, the exact measures of angles 1 and 2 cannot be uniquely determined from the given information. Additional information would be required to solve for each angle individually.
### Conclusion:
By following the steps above, we utilized geometric properties to determine the relationship between the angles in the hexagon. Using the provided measurements, we calculated the total degree of the unknown angles which sums to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ac7b839-49d7-494c-ab52-df226dbddc3d%2F5520b121-61f5-48f6-9718-eec77a3acf5d%2Funzlt1h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Measures of Angles in a Hexagon
**Problem Statement:**
Given the hexagon below, find the measures of angles 1 through 7.
[In the image provided, a hexagon is presented with various angle measurements labeled.]
**Measurements and Internal Angles:**
1. Angle 3: 57°
2. Angle 4: 118°
3. Angle 5: 64°
4. Angle 6: 164°
5. Angle 7: 88°
**Steps for Finding Unknown Angles:**
1. **Hexagon Angle Sum Property:**
- The sum of the interior angles of a hexagon is given by the formula:
\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]
where \( n \) is the number of sides. For a hexagon ( \( n = 6 \) ):
\[
(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]
2. **Using Given Angles to Find Missing Angles:**
- Known angles are given as:
\[
57°, 118°, 64°, 164°, 88°
\]
- Calculate the sum of these known angles:
\[
57° + 118° + 64° + 164° + 88° = 491°
\]
3. **Finding the Remaining Angles (1 and 2):**
- Let angles 1 and 2 be \( x \) and \( y \) respectively. The sum of these angles can be found as:
\[
x + y + 491° = 720°
\]
\[
x + y = 720° - 491° = 229°
\]
- Since the relationship between \( x \) and \( y \) is not further defined in the problem, the exact measures of angles 1 and 2 cannot be uniquely determined from the given information. Additional information would be required to solve for each angle individually.
### Conclusion:
By following the steps above, we utilized geometric properties to determine the relationship between the angles in the hexagon. Using the provided measurements, we calculated the total degree of the unknown angles which sums to
![**Finding the Area of a Shaded Segment in a Circle**
To determine the area of the shaded segment in the given circle, follow these steps:
### Diagram Explanation
1. **Circle**: The figure shows a circle with center \( O \).
2. **Segments**: Line segments \( OM \) and \( ON \) represent radii of the circle, each measuring 12 inches.
3. **Chord and Central Angle**: Line segment \( MN \) is a chord in the circle. The angle \(\angle MON\) formed by the radii \( OM \) and \( ON \) is \( 30^\circ \).
4. **Shaded Segment**: The area that needs to be found is the shaded segment \( MPN \), which lies outside the triangle \( OMN \) but within the circular sector bounded by arc \( MN \).
### Step-by-Step Solution
1. **Calculate the Area of Sector \( OMON \)**:
- The entire circle's area is \( \pi r^2 = \pi (12)^2 = 144\pi \) square inches.
- The area of a sector with a central angle of \( \theta \) is given by \( \frac{\theta}{360^\circ} \times \pi r^2 \).
- For \( \angle MON = 30^\circ \):
\[
\text{Area of Sector} = \left(\frac{30}{360}\right) \times 144\pi = \frac{1}{12} \times 144\pi = 12\pi \, \text{square inches}
\]
2. **Calculate the Area of Triangle \( OMN \)**:
- With \( OM \) and \( ON \) as equal radii (12 inches) and \( \angle MON = 30^\circ \):
- The area of triangle \( \triangle OMN \) can be computed using the formula \( \frac{1}{2}ab \sin(C) \):
\[
\text{Area of } \triangle OMN = \frac{1}{2} \cdot 12 \cdot 12 \cdot \sin(30^\circ) = \frac{1}{2} \cdot 12 \cdot 12 \cdot \frac{1}{2} = 36 \, \text{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ac7b839-49d7-494c-ab52-df226dbddc3d%2F5520b121-61f5-48f6-9718-eec77a3acf5d%2F2ddtaxb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Area of a Shaded Segment in a Circle**
To determine the area of the shaded segment in the given circle, follow these steps:
### Diagram Explanation
1. **Circle**: The figure shows a circle with center \( O \).
2. **Segments**: Line segments \( OM \) and \( ON \) represent radii of the circle, each measuring 12 inches.
3. **Chord and Central Angle**: Line segment \( MN \) is a chord in the circle. The angle \(\angle MON\) formed by the radii \( OM \) and \( ON \) is \( 30^\circ \).
4. **Shaded Segment**: The area that needs to be found is the shaded segment \( MPN \), which lies outside the triangle \( OMN \) but within the circular sector bounded by arc \( MN \).
### Step-by-Step Solution
1. **Calculate the Area of Sector \( OMON \)**:
- The entire circle's area is \( \pi r^2 = \pi (12)^2 = 144\pi \) square inches.
- The area of a sector with a central angle of \( \theta \) is given by \( \frac{\theta}{360^\circ} \times \pi r^2 \).
- For \( \angle MON = 30^\circ \):
\[
\text{Area of Sector} = \left(\frac{30}{360}\right) \times 144\pi = \frac{1}{12} \times 144\pi = 12\pi \, \text{square inches}
\]
2. **Calculate the Area of Triangle \( OMN \)**:
- With \( OM \) and \( ON \) as equal radii (12 inches) and \( \angle MON = 30^\circ \):
- The area of triangle \( \triangle OMN \) can be computed using the formula \( \frac{1}{2}ab \sin(C) \):
\[
\text{Area of } \triangle OMN = \frac{1}{2} \cdot 12 \cdot 12 \cdot \sin(30^\circ) = \frac{1}{2} \cdot 12 \cdot 12 \cdot \frac{1}{2} = 36 \, \text{
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