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5х - 8 2x +13 P'

5х - 8 2x +13 P'

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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please help me answer questions 6 and 14

### Geometry Problem 6 The diagram presents a geometric figure consisting of two adjacent right triangles, labeled \( \triangle GMK \) and \( \triangle GMP \). Both triangles share the vertex \( G \) and the side \( GM \). - **Vertices**: The vertices of the figure are labeled as \( G \), \( K \), \( M \), and \( P \). - **Angles**: - The angle at vertex \( M \) in \( \triangle GMK \) is a right angle (90 degrees), and is marked with a small square. - Similarly, the angle at vertex \( M \) in \( \triangle GMP \) is also a right angle (90 degrees), indicated by a small square. - **Angle Measures**: - The external angle at vertex \( G \) between sides \( GK \) and \( GM \) is labeled \( 5x - 8 \). - The external angle at vertex \( G \) between sides \( GM \) and \( GP \) is labeled \( 2x + 13 \). This geometric representation is commonly used to illustrate the concepts of right angle triangles, angle relationships, and can be used in problems involving algebraic expressions and geometric properties. For analysis: - Consider the relationships between the angles in adjacent triangles and the properties of right triangles. - Utilize algebraic methods to solve for the variable \( x \) if needed. Note: Ensure that the sum of the angles around point \( G \) equals \( 360^\circ \), a fundamental property in planar geometry.
Transcribed Image Text:### Geometry Problem 6 The diagram presents a geometric figure consisting of two adjacent right triangles, labeled \( \triangle GMK \) and \( \triangle GMP \). Both triangles share the vertex \( G \) and the side \( GM \). - **Vertices**: The vertices of the figure are labeled as \( G \), \( K \), \( M \), and \( P \). - **Angles**: - The angle at vertex \( M \) in \( \triangle GMK \) is a right angle (90 degrees), and is marked with a small square. - Similarly, the angle at vertex \( M \) in \( \triangle GMP \) is also a right angle (90 degrees), indicated by a small square. - **Angle Measures**: - The external angle at vertex \( G \) between sides \( GK \) and \( GM \) is labeled \( 5x - 8 \). - The external angle at vertex \( G \) between sides \( GM \) and \( GP \) is labeled \( 2x + 13 \). This geometric representation is commonly used to illustrate the concepts of right angle triangles, angle relationships, and can be used in problems involving algebraic expressions and geometric properties. For analysis: - Consider the relationships between the angles in adjacent triangles and the properties of right triangles. - Utilize algebraic methods to solve for the variable \( x \) if needed. Note: Ensure that the sum of the angles around point \( G \) equals \( 360^\circ \), a fundamental property in planar geometry.
### Problem 14. **Given:** An isosceles triangle is shown with the two equal sides marked with a short line each. The angles opposite those sides are marked as \( x^\circ \) (at the top vertex) and \( y^\circ \) (at the bottom right vertex). The angle at the bottom left vertex is given as \( 43^\circ \). **Find:** Values of \( x \) and \( y \). **Solution:** 1. In an isosceles triangle, the angles opposite to the equal sides are also equal. Hence, \( x = y \). 2. The sum of all interior angles in any triangle is always \( 180^\circ \). Therefore: \[ x + y + 43^\circ = 180^\circ \] 3. Since \( x = y \): \[ x + x + 43^\circ = 180^\circ \] \[ 2x + 43^\circ = 180^\circ \] 4. Solving for \( x \): \[ 2x = 180^\circ - 43^\circ \] \[ 2x = 137^\circ \] \[ x = \frac{137^\circ}{2} \] \[ x = 68.5^\circ \] 5. Since \( x = y \): \[ y = 68.5^\circ \] **Conclusion:** \[ x = 68.5^\circ \quad y = 68.5^\circ \] **Answer:** \[ x = 68.5^\circ \quad y = 68.5^\circ \]
Transcribed Image Text:### Problem 14. **Given:** An isosceles triangle is shown with the two equal sides marked with a short line each. The angles opposite those sides are marked as \( x^\circ \) (at the top vertex) and \( y^\circ \) (at the bottom right vertex). The angle at the bottom left vertex is given as \( 43^\circ \). **Find:** Values of \( x \) and \( y \). **Solution:** 1. In an isosceles triangle, the angles opposite to the equal sides are also equal. Hence, \( x = y \). 2. The sum of all interior angles in any triangle is always \( 180^\circ \). Therefore: \[ x + y + 43^\circ = 180^\circ \] 3. Since \( x = y \): \[ x + x + 43^\circ = 180^\circ \] \[ 2x + 43^\circ = 180^\circ \] 4. Solving for \( x \): \[ 2x = 180^\circ - 43^\circ \] \[ 2x = 137^\circ \] \[ x = \frac{137^\circ}{2} \] \[ x = 68.5^\circ \] 5. Since \( x = y \): \[ y = 68.5^\circ \] **Conclusion:** \[ x = 68.5^\circ \quad y = 68.5^\circ \] **Answer:** \[ x = 68.5^\circ \quad y = 68.5^\circ \]
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