Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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**Topic: Solving for Angles when Given Parallel Lines** **Given Parallel Lines** When you are given that two lines are parallel, many different geometric properties can be used to find missing angles. In this example, we'll use these properties to find the value of \( x \). **Problem Statement:** Given that \( m \parallel n \), find the value of \( x \). **Diagram Explanation:** The diagram shows two parallel lines, \( m \) and \( n \), intersected by a transversal. Two angles are given in the diagram: 1. An angle \( (x + 27)^\circ \) on the upper right side of the transversal. 2. An angle \( (2x - 7)^\circ \) on the lower right side of the transversal. **Geometric Properties to Use:** When two parallel lines are intersected by a transversal: 1. Corresponding Angles are equal. 2. Alternate Interior Angles are equal. 3. Consecutive Interior Angles are supplementary (sum up to \( 180^\circ \)). **Steps to Solve:** 1. Identify the relationship between the given angles. - Here, since \( (x + 27)^\circ \) and \( (2x - 7)^\circ \) are formed by parallel lines intersected by a transversal, they must be related by one of the properties above. In this scenario, they are Consecutive Interior Angles. 2. Set up the equation based on the consecutive interior angles property. - \((x + 27) + (2x - 7) = 180\) 3. Simplify and solve for \( x \). - \(x + 27 + 2x - 7 = 180\) - \(3x + 20 = 180\) - \(3x = 160\) - \(x = \frac{160}{3}\) - \(x = 53.33\) **Conclusion:** The value of \( x \) in this problem is \( 53.33 \). This method can be used to find missing angles whenever you have parallel lines and a transversal, by applying the properties of corresponding, alternate interior, or consecutive interior angles.
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Transcribed Image Text:**Topic: Solving for Angles when Given Parallel Lines** **Given Parallel Lines** When you are given that two lines are parallel, many different geometric properties can be used to find missing angles. In this example, we'll use these properties to find the value of \( x \). **Problem Statement:** Given that \( m \parallel n \), find the value of \( x \). **Diagram Explanation:** The diagram shows two parallel lines, \( m \) and \( n \), intersected by a transversal. Two angles are given in the diagram: 1. An angle \( (x + 27)^\circ \) on the upper right side of the transversal. 2. An angle \( (2x - 7)^\circ \) on the lower right side of the transversal. **Geometric Properties to Use:** When two parallel lines are intersected by a transversal: 1. Corresponding Angles are equal. 2. Alternate Interior Angles are equal. 3. Consecutive Interior Angles are supplementary (sum up to \( 180^\circ \)). **Steps to Solve:** 1. Identify the relationship between the given angles. - Here, since \( (x + 27)^\circ \) and \( (2x - 7)^\circ \) are formed by parallel lines intersected by a transversal, they must be related by one of the properties above. In this scenario, they are Consecutive Interior Angles. 2. Set up the equation based on the consecutive interior angles property. - \((x + 27) + (2x - 7) = 180\) 3. Simplify and solve for \( x \). - \(x + 27 + 2x - 7 = 180\) - \(3x + 20 = 180\) - \(3x = 160\) - \(x = \frac{160}{3}\) - \(x = 53.33\) **Conclusion:** The value of \( x \) in this problem is \( 53.33 \). This method can be used to find missing angles whenever you have parallel lines and a transversal, by applying the properties of corresponding, alternate interior, or consecutive interior angles.
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