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6. 10 899 y° 15 30

6. 10 899 y° 15 30

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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Shown are two similar polygons. Find the values of x, y, and z. 

The image features two right-angled triangles with given side lengths and angles. Here's the detailed transcription: ### Right-Angled Triangles #### Triangle 1 (Left) - **Vertices:** Not labeled with specific letters. - **Known Sides:** - One side adjacent to the 89° angle is labeled as 6. - The opposite side (base of the right angle) is labeled as 15. - **Known Angle:** - The angle opposite to the side labeled 6 is 89°. - **Unknowns:** - The hypothenuse, labeled as \( z \). - The smallest angle at the bottom left corner, opposite to the side labeled 15, is not explicitly given. #### Triangle 2 (Right) - **Vertices:** Not labeled with specific letters. - **Known Sides:** - One side adjacent to the unknown angle \( y \) is labeled as 10. - The opposite side (base of the right angle) is labeled as 30. - **Known Angle:** - The second known angle is represented as \( y° \). - **Unknowns:** - The hypothenuse, labeled as \( x \). - The unknown angle \( y°\). ### Diagram Description - **Right-Angled Triangle Configuration:** - Both triangles have one right angle each, indicated by the small square at the corner. - The side lengths and angles are marked within the triangles. - **Angle Indicator:** - The angles opposite to the marked sides are indicated in degrees. Before delving into solving for the unknowns, let's summarize possible steps for educational purposes: - **Pythagorean Theorem** can be used to find the length of the hypotenuse (e.g., \( c = \sqrt{a^2 + b^2} \)). - Trigonometric functions such as **sine, cosine, and tangent** can help find unknown angles and sides (e.g., \( \sin \theta = \frac{opposite}{hypotenuse} \)). ### Mathematical Formulas 1. **Pythagorean Theorem:** - \( \text{Hypotenuse}^2 = \text{Adjacent side}^2 + \text{Opposite side}^2 \) 2. **Trigonometric Ratios:** - \( \sin(\
Transcribed Image Text:The image features two right-angled triangles with given side lengths and angles. Here's the detailed transcription: ### Right-Angled Triangles #### Triangle 1 (Left) - **Vertices:** Not labeled with specific letters. - **Known Sides:** - One side adjacent to the 89° angle is labeled as 6. - The opposite side (base of the right angle) is labeled as 15. - **Known Angle:** - The angle opposite to the side labeled 6 is 89°. - **Unknowns:** - The hypothenuse, labeled as \( z \). - The smallest angle at the bottom left corner, opposite to the side labeled 15, is not explicitly given. #### Triangle 2 (Right) - **Vertices:** Not labeled with specific letters. - **Known Sides:** - One side adjacent to the unknown angle \( y \) is labeled as 10. - The opposite side (base of the right angle) is labeled as 30. - **Known Angle:** - The second known angle is represented as \( y° \). - **Unknowns:** - The hypothenuse, labeled as \( x \). - The unknown angle \( y°\). ### Diagram Description - **Right-Angled Triangle Configuration:** - Both triangles have one right angle each, indicated by the small square at the corner. - The side lengths and angles are marked within the triangles. - **Angle Indicator:** - The angles opposite to the marked sides are indicated in degrees. Before delving into solving for the unknowns, let's summarize possible steps for educational purposes: - **Pythagorean Theorem** can be used to find the length of the hypotenuse (e.g., \( c = \sqrt{a^2 + b^2} \)). - Trigonometric functions such as **sine, cosine, and tangent** can help find unknown angles and sides (e.g., \( \sin \theta = \frac{opposite}{hypotenuse} \)). ### Mathematical Formulas 1. **Pythagorean Theorem:** - \( \text{Hypotenuse}^2 = \text{Adjacent side}^2 + \text{Opposite side}^2 \) 2. **Trigonometric Ratios:** - \( \sin(\
### Understanding Trapezoids This image contains two geometric shapes, specifically trapezoids, each with various measurements. #### Trapezoid on the Left: This trapezoid has the following characteristics: - One of its angles is denoted as \( y^\circ \). - Another angle is denoted as \( z^\circ \). - The left side of the trapezoid adjacent to \( y^\circ \) measures \( 4 \) units. - The top side of the trapezoid opposite \( z^\circ \) is represented as \( x \) units. #### Trapezoid on the Right: This trapezoid includes: - Two right angles, each measuring \( 90^\circ \). - The top side measures \( 8 \) units. - The bottom side measures \( 24 \) units. - The angle between the slanted side and the bottom side of the trapezoid measures \( 38^\circ \). ### Key Concepts: - **Angles**: - \( y^\circ \) and \( z^\circ \) are internal angles of the left trapezoid. - \( 38^\circ \), \( 90^\circ \), and \( 90^\circ \) are internal angles of the right trapezoid. - **Parallel Sides**: - In trapezoids, one pair of opposite sides is parallel. The sides of the trapezoids labeled \( x \) and \( 4 \), and those labeled \( 8 \) and \( 24 \) represent these parallel sides. - **Right Angles**: - Right angles are \( 90^\circ \). The right trapezoid has two right angles, which are indicated in the illustration. ### Applying Geometry Theorems: #### For the Left Trapezoid: - If \( y \) and \( z \) are the internal angles in a trapezoid, they must sum to \( 180^\circ \) if the non-parallel sides are an extension of the same line. #### For the Right Trapezoid: - The sum of all angles inside a trapezoid is always \( 360^\circ \). Given two right angles and one angle of \( 38^\circ \), the remaining angle can be calculated as follows: - \( 90^\circ + 90^\circ + 38^\circ \)
Transcribed Image Text:### Understanding Trapezoids This image contains two geometric shapes, specifically trapezoids, each with various measurements. #### Trapezoid on the Left: This trapezoid has the following characteristics: - One of its angles is denoted as \( y^\circ \). - Another angle is denoted as \( z^\circ \). - The left side of the trapezoid adjacent to \( y^\circ \) measures \( 4 \) units. - The top side of the trapezoid opposite \( z^\circ \) is represented as \( x \) units. #### Trapezoid on the Right: This trapezoid includes: - Two right angles, each measuring \( 90^\circ \). - The top side measures \( 8 \) units. - The bottom side measures \( 24 \) units. - The angle between the slanted side and the bottom side of the trapezoid measures \( 38^\circ \). ### Key Concepts: - **Angles**: - \( y^\circ \) and \( z^\circ \) are internal angles of the left trapezoid. - \( 38^\circ \), \( 90^\circ \), and \( 90^\circ \) are internal angles of the right trapezoid. - **Parallel Sides**: - In trapezoids, one pair of opposite sides is parallel. The sides of the trapezoids labeled \( x \) and \( 4 \), and those labeled \( 8 \) and \( 24 \) represent these parallel sides. - **Right Angles**: - Right angles are \( 90^\circ \). The right trapezoid has two right angles, which are indicated in the illustration. ### Applying Geometry Theorems: #### For the Left Trapezoid: - If \( y \) and \( z \) are the internal angles in a trapezoid, they must sum to \( 180^\circ \) if the non-parallel sides are an extension of the same line. #### For the Right Trapezoid: - The sum of all angles inside a trapezoid is always \( 360^\circ \). Given two right angles and one angle of \( 38^\circ \), the remaining angle can be calculated as follows: - \( 90^\circ + 90^\circ + 38^\circ \)
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