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хо 143⁰ 120° 137⁰ 115° 90°

хо 143⁰ 120° 137⁰ 115° 90°

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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**Understanding Interior Angles in a Polygon** In this illustration, we have a polygon with six sides, also referred to as a hexagon. Each interior angle of the polygon is marked with its respective degree measurement. ### Interior Angles of the Polygon: 1. **First Interior Angle:** 137° 2. **Second Interior Angle:** 90° 3. **Third Interior Angle:** 115° 4. **Fourth Interior Angle:** 120° 5. **Fifth Interior Angle:** 143° 6. **Sixth Interior Angle:** \( x \) ### Explanation of the Diagram: - The hexagon has six vertices where each interior angle is placed. - Five of these angles are explicitly marked with their degree measurements. - The sixth interior angle is denoted as \( x \), representing an unknown value that can be calculated. ### Calculating the Unknown Angle: To find the unknown angle \( x \), we use the property of polygons that states the sum of interior angles of a polygon with \( n \) sides is given by the formula: \[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \] For a hexagon (\( n = 6 \)): \[ \text{Sum of interior angles} = (6 - 2) \times 180^\circ \] \[ = 4 \times 180^\circ \] \[ = 720^\circ \] Now, adding the given angles and subtracting from the total sum will give us the unknown angle \( x \): \[ 137^\circ + 90^\circ + 115^\circ + 120^\circ + 143^\circ + x = 720^\circ \] Combining the known angles: \[ 137^\circ + 90^\circ + 115^\circ + 120^\circ + 143^\circ = 605^\circ \] Therefore, solving for \( x \): \[ 605^\circ + x = 720^\circ \] \[ x = 720^\circ - 605^\circ \] \[ x = 115^\circ \] Thus, the unknown angle \( x \) is \( 115^\circ \). This exercise is an excellent way to apply the formula for the sum of interior angles of a polygon and understand the relationship between the angles inside a hexagon.
Transcribed Image Text:**Understanding Interior Angles in a Polygon** In this illustration, we have a polygon with six sides, also referred to as a hexagon. Each interior angle of the polygon is marked with its respective degree measurement. ### Interior Angles of the Polygon: 1. **First Interior Angle:** 137° 2. **Second Interior Angle:** 90° 3. **Third Interior Angle:** 115° 4. **Fourth Interior Angle:** 120° 5. **Fifth Interior Angle:** 143° 6. **Sixth Interior Angle:** \( x \) ### Explanation of the Diagram: - The hexagon has six vertices where each interior angle is placed. - Five of these angles are explicitly marked with their degree measurements. - The sixth interior angle is denoted as \( x \), representing an unknown value that can be calculated. ### Calculating the Unknown Angle: To find the unknown angle \( x \), we use the property of polygons that states the sum of interior angles of a polygon with \( n \) sides is given by the formula: \[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \] For a hexagon (\( n = 6 \)): \[ \text{Sum of interior angles} = (6 - 2) \times 180^\circ \] \[ = 4 \times 180^\circ \] \[ = 720^\circ \] Now, adding the given angles and subtracting from the total sum will give us the unknown angle \( x \): \[ 137^\circ + 90^\circ + 115^\circ + 120^\circ + 143^\circ + x = 720^\circ \] Combining the known angles: \[ 137^\circ + 90^\circ + 115^\circ + 120^\circ + 143^\circ = 605^\circ \] Therefore, solving for \( x \): \[ 605^\circ + x = 720^\circ \] \[ x = 720^\circ - 605^\circ \] \[ x = 115^\circ \] Thus, the unknown angle \( x \) is \( 115^\circ \). This exercise is an excellent way to apply the formula for the sum of interior angles of a polygon and understand the relationship between the angles inside a hexagon.
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