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X 89 □ P 37² N

X 89 □ P 37² N

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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**Title: Solving for the Missing Angle in a Right Triangle** **Objective:** Learn how to solve for the unknown angle in a right-angled triangle. **Diagram Description:** The diagram provided illustrates a right-angled triangle labeled PAXN. Here's a detailed breakdown of the triangle's components: - **Vertices:** The vertices are labeled as P, A, and N. - **Right Angle:** The right angle (90°) is at vertex A. - **Angle PAX:** This angle measures 89 degrees. - **Angle PAN:** This angle measures 37 degrees. The sides opposite and adjacent to these angles are not labeled with side lengths. **Problem:** Solve for the missing angle at vertex X. **Solution Steps:** 1. **Identify Known Angles:** - Angle PAX = 89° - Angle PAN = 37° - Angle at A (right angle) = 90° 2. **Calculate the Missing Angle (∠XAN):** Since the sum of angles in any triangle is always 180 degrees: \[ 90° (Angle at A) + 37° (PAN) + ∠XAN = 180° \] 3. **Sum the Known Angles:** \[ 90° + 37° = 127° \] 4. **Determine the Missing Angle:** \[ ∠XAN = 180° - 127° \] \[ ∠XAN = 53° \] Therefore, the missing angle ∠XAN is 53 degrees. **Conclusion:** By understanding the properties of triangles and using basic angle sum properties, we've determined that the missing angle ∠XAN in the right triangle PAXN is 53 degrees.
Transcribed Image Text:**Title: Solving for the Missing Angle in a Right Triangle** **Objective:** Learn how to solve for the unknown angle in a right-angled triangle. **Diagram Description:** The diagram provided illustrates a right-angled triangle labeled PAXN. Here's a detailed breakdown of the triangle's components: - **Vertices:** The vertices are labeled as P, A, and N. - **Right Angle:** The right angle (90°) is at vertex A. - **Angle PAX:** This angle measures 89 degrees. - **Angle PAN:** This angle measures 37 degrees. The sides opposite and adjacent to these angles are not labeled with side lengths. **Problem:** Solve for the missing angle at vertex X. **Solution Steps:** 1. **Identify Known Angles:** - Angle PAX = 89° - Angle PAN = 37° - Angle at A (right angle) = 90° 2. **Calculate the Missing Angle (∠XAN):** Since the sum of angles in any triangle is always 180 degrees: \[ 90° (Angle at A) + 37° (PAN) + ∠XAN = 180° \] 3. **Sum the Known Angles:** \[ 90° + 37° = 127° \] 4. **Determine the Missing Angle:** \[ ∠XAN = 180° - 127° \] \[ ∠XAN = 53° \] Therefore, the missing angle ∠XAN is 53 degrees. **Conclusion:** By understanding the properties of triangles and using basic angle sum properties, we've determined that the missing angle ∠XAN in the right triangle PAXN is 53 degrees.
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