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 Formulas1. **Pythagorean Theorem:** - $ \text{Hypotenuse}^2 = \text{Adjacent side}^2 + \text{Opposite side}^2 $2. **Trigonometric Ratios:** - $ \sin(\ ### Understanding TrapezoidsThis 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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Answered: 6. 10 899 y° 15 30

Math

Geometry

6. 10 899 y° 15 30

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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Question

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(\

### 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 $

Definition Definition Two-dimentional plane figure composed of a finite number of straight line segments connected to form a closed chain or circuit. A polygonal circuit's segments are known as its edges or sides, and the points where two edges meet are known as its vertices or corners.

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