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24. _b° 44° 160°

24. _b° 44° 160°

Elementary Geometry For College Students, 7e
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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Show work and then lastly describe your answers in 3 sentences. They are two questions ( parts of 1 question please answer separate tho) 

 

18. Find the value of each variable. Lines that appear to be tangent are tangent. 

24. Find the value of each variable. For each circle, the dot represents the center. 

 

 

**Problem 18:** In the given diagram, we have two circles tangent to each other and to a straight line. The first circle is larger and has a marked angle of \(115^\circ\) inside it. The second, smaller circle forms an angle \(e^\circ\) at the intersection with the straight tangent line, and there is another angle \(f^\circ\) at the intersection of the smaller circle and the tangent line. The task is to determine the relationships between the given angles. 1. **First Circle (Larger Circle):** - The angle marked within the larger circle is \(115^\circ\). 2. **Second Circle (Smaller Circle):** - Two angles, \(e^\circ\) and \(f^\circ\), are formed at the points where the smaller circle touches the tangent line. To solve for the angles \(e^\circ\) and \(f^\circ\), certain geometric properties and theorems such as the tangent-secant theorem, properties of angles in circles, and the fact that the tangent to a circle forms a right angle with the radius at the point of tangency need to be considered. **Explanation of Angles and Circle Properties:** - **Angles in a Circle:** - Any angle inscribed in a semicircle is a right angle (90 degrees). - The sum of angles around a point is \(360^\circ\). - The tangent to a circle makes a right angle with the radius at the point of tangency. Given that the external line is a tangent and the right angle formed will help understand the complementary and supplementary angle relationships to solve for \(e^\circ\) and \(f^\circ\). Having the provided measures and relationships, proceed to find \(e^\circ\) and \(f^\circ\) using geometric principles relevant to the structure and layout of the diagram.
Transcribed Image Text:**Problem 18:** In the given diagram, we have two circles tangent to each other and to a straight line. The first circle is larger and has a marked angle of \(115^\circ\) inside it. The second, smaller circle forms an angle \(e^\circ\) at the intersection with the straight tangent line, and there is another angle \(f^\circ\) at the intersection of the smaller circle and the tangent line. The task is to determine the relationships between the given angles. 1. **First Circle (Larger Circle):** - The angle marked within the larger circle is \(115^\circ\). 2. **Second Circle (Smaller Circle):** - Two angles, \(e^\circ\) and \(f^\circ\), are formed at the points where the smaller circle touches the tangent line. To solve for the angles \(e^\circ\) and \(f^\circ\), certain geometric properties and theorems such as the tangent-secant theorem, properties of angles in circles, and the fact that the tangent to a circle forms a right angle with the radius at the point of tangency need to be considered. **Explanation of Angles and Circle Properties:** - **Angles in a Circle:** - Any angle inscribed in a semicircle is a right angle (90 degrees). - The sum of angles around a point is \(360^\circ\). - The tangent to a circle makes a right angle with the radius at the point of tangency. Given that the external line is a tangent and the right angle formed will help understand the complementary and supplementary angle relationships to solve for \(e^\circ\) and \(f^\circ\). Having the provided measures and relationships, proceed to find \(e^\circ\) and \(f^\circ\) using geometric principles relevant to the structure and layout of the diagram.
### Geometry Problem: Angles in a Circle #### Problem Statement: Given a circle with a triangle inscribed within it, the following angles are provided: - An interior angle within the triangle is 44°. - One of the angles outside the triangle but subtending the same arc as the triangle is 160°. - The angle \(a^\circ\) lies outside the circle and forms the complementary angle to one side of the triangle. - The angle \(b^\circ\) is the exterior angle adjacent to \(a^\circ\). - The angle \(c^\circ\) is formed by the arc opposite the interior angle of 44°. #### Diagram Explanation: 1. A circle with a triangle inscribed. 2. The interior angle of the triangle within the circle is 44°. 3. An angle outside the circle (but subtending the same arc as the interior angle) is given as 160°. 4. The angles \(a^\circ\), \(b^\circ\), and \(c^\circ\) are marked on the figure, where: - \(a^\circ\) is an angle formed outside the circle. - \(b^\circ\) is adjacent to \(a^\circ\). - \(c^\circ\) is the angle subtended by the arc opposite the 44° interior angle. #### Key Concepts: 1. **Central Angle Theorem**: The angle \(\angle c^\circ\) at the center of the circle is twice the angle subtended at the circumference (44°), which means \(\angle c = 88°\). 2. **Exterior Angles**: The exterior angle \(b^\circ\) plus angle \(a^\circ\) equals 160°. **Note**: Ensure to use these angles appropriately to solve the geometric relations and properties derived from the circle theorems.
Transcribed Image Text:### Geometry Problem: Angles in a Circle #### Problem Statement: Given a circle with a triangle inscribed within it, the following angles are provided: - An interior angle within the triangle is 44°. - One of the angles outside the triangle but subtending the same arc as the triangle is 160°. - The angle \(a^\circ\) lies outside the circle and forms the complementary angle to one side of the triangle. - The angle \(b^\circ\) is the exterior angle adjacent to \(a^\circ\). - The angle \(c^\circ\) is formed by the arc opposite the interior angle of 44°. #### Diagram Explanation: 1. A circle with a triangle inscribed. 2. The interior angle of the triangle within the circle is 44°. 3. An angle outside the circle (but subtending the same arc as the interior angle) is given as 160°. 4. The angles \(a^\circ\), \(b^\circ\), and \(c^\circ\) are marked on the figure, where: - \(a^\circ\) is an angle formed outside the circle. - \(b^\circ\) is adjacent to \(a^\circ\). - \(c^\circ\) is the angle subtended by the arc opposite the 44° interior angle. #### Key Concepts: 1. **Central Angle Theorem**: The angle \(\angle c^\circ\) at the center of the circle is twice the angle subtended at the circumference (44°), which means \(\angle c = 88°\). 2. **Exterior Angles**: The exterior angle \(b^\circ\) plus angle \(a^\circ\) equals 160°. **Note**: Ensure to use these angles appropriately to solve the geometric relations and properties derived from the circle theorems.
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