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To identify the most efficient model for a windmill, Gandalf Engineering Company designs the windmill shown. The design consists of four right triangles arranged equally around point O such that point O is the midpoint of BF and DH. Use deductive reasoning to prove the following: Given: Point O is the midpoint of BF. AABO & AEFO are right triangles with right angles ZB & ZF.

To identify the most efficient model for a windmill, Gandalf Engineering Company designs the windmill shown. The design consists of four right triangles arranged equally around point O such that point O is the midpoint of BF and DH. Use deductive reasoning to prove the following: Given: Point O is the midpoint of BF. AABO & AEFO are right triangles with right angles ZB & ZF.

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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To identify the most efficient model for a windmill, Gandalf Engineering Company designs the windmill shown. The design consists of four right triangles arranged equally around point O such that point O is the midpoint of \( \overline{BF} \) and \( \overline{DH} \). ![Windmill Diagram](windmill.jpg) The diagram shows four right triangles colored pink, orange, purple, and blue, arranged around a central point O. These triangles are labeled as follows: - \( \triangle ABO \) - \( \triangle CDO \) - \( \triangle EFO \) - \( \triangle GHO \) The right angles are at points \( A, C, E,\) and \(G\), with their hypotenuses converging at point O. Use deductive reasoning to prove the following: **Given:** Point \( O \) is the midpoint of \( \overline{BF} \). \( \triangle ABO \) and \( \triangle EFO \) are right triangles with right angles \( \angle B \) and \( \angle F \). **Prove:** \( \angle A \cong \angle E \) **Steps to prove:** 1. **Identify given information:** - Point \( O \) is the midpoint of \( \overline{BF} \). - Right triangles \( \triangle ABO \) and \( \triangle EFO \) have right angles at \( \angle B \) and \( \angle F \). 2. **Use properties of midpoint and right triangles:** - Since \( O \) is the midpoint, we have that \( \overline{BO} \cong \overline{OF} \). - \( \angle B \) and \( \angle F \) are right angles, therefore \( \triangle ABO \) and \( \triangle EFO \) are similar by AA (angle-angle) similarity. 3. **Use congruence to prove angles:** - Because the triangles are similar, the corresponding angles are congruent. - Therefore, \( \angle A \cong \angle E \). This concludes the proof that \( \angle A \cong \angle E \).
Transcribed Image Text:To identify the most efficient model for a windmill, Gandalf Engineering Company designs the windmill shown. The design consists of four right triangles arranged equally around point O such that point O is the midpoint of \( \overline{BF} \) and \( \overline{DH} \). ![Windmill Diagram](windmill.jpg) The diagram shows four right triangles colored pink, orange, purple, and blue, arranged around a central point O. These triangles are labeled as follows: - \( \triangle ABO \) - \( \triangle CDO \) - \( \triangle EFO \) - \( \triangle GHO \) The right angles are at points \( A, C, E,\) and \(G\), with their hypotenuses converging at point O. Use deductive reasoning to prove the following: **Given:** Point \( O \) is the midpoint of \( \overline{BF} \). \( \triangle ABO \) and \( \triangle EFO \) are right triangles with right angles \( \angle B \) and \( \angle F \). **Prove:** \( \angle A \cong \angle E \) **Steps to prove:** 1. **Identify given information:** - Point \( O \) is the midpoint of \( \overline{BF} \). - Right triangles \( \triangle ABO \) and \( \triangle EFO \) have right angles at \( \angle B \) and \( \angle F \). 2. **Use properties of midpoint and right triangles:** - Since \( O \) is the midpoint, we have that \( \overline{BO} \cong \overline{OF} \). - \( \angle B \) and \( \angle F \) are right angles, therefore \( \triangle ABO \) and \( \triangle EFO \) are similar by AA (angle-angle) similarity. 3. **Use congruence to prove angles:** - Because the triangles are similar, the corresponding angles are congruent. - Therefore, \( \angle A \cong \angle E \). This concludes the proof that \( \angle A \cong \angle E \).
### Proving ∠A ≈ ∠E #### Given: - Point O is the midpoint of \( \overline{BF} \). - \( \triangle ABO \) and \( \triangle EFO \) are right triangles with right angles \( \angle B \) and \( \angle F \). #### To Prove: - \( \angle A \cong \angle E \). #### Proof Table: | Statements | Justifications | |---------------------------------------------------------|---------------------------------------| | 1. Point O is the midpoint of \( \overline{BF} \). \( \triangle ABO \) and \( \triangle EFO \) are right triangles with right angles \( \angle B \) and \( \angle F \). | 1. Given | | 2. \( \overline{BO} \cong \overline{FO} \) | 2. Definition of midpoint | | 3. \( \angle B \cong \angle F \) | 3. Right Angle Congruence Theorem | | 4. \( \overline{AO} \cong \overline{EO} \) | 4. Given | | 5. \( \triangle ABO \cong \triangle EFO \) | 5. SAS | | 6. \( \angle A \cong \angle E \) | 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) | #### Explanation of Diagrams and Proofs: The proof involves showing that two right triangles \( \triangle ABO \) and \( \triangle EFO \) are congruent based on the SAS (Side-Angle-Side) criteria. To do this, several key statements and justifications are employed: 1. ***Point O is identified as the midpoint***, setting up the basis for why \( \overline{BO} \) is congruent to \( \overline{FO} \). 2. The fact that ***\( \angle B \) and \( \angle F \) are right angles*** confirms their congruence via the Right Angle Congruence Theorem. 3. ***\( \overline{AO} \) and \( \overline{EO} \) being congruent*** is given, as these are corresponding sides in the right
Transcribed Image Text:### Proving ∠A ≈ ∠E #### Given: - Point O is the midpoint of \( \overline{BF} \). - \( \triangle ABO \) and \( \triangle EFO \) are right triangles with right angles \( \angle B \) and \( \angle F \). #### To Prove: - \( \angle A \cong \angle E \). #### Proof Table: | Statements | Justifications | |---------------------------------------------------------|---------------------------------------| | 1. Point O is the midpoint of \( \overline{BF} \). \( \triangle ABO \) and \( \triangle EFO \) are right triangles with right angles \( \angle B \) and \( \angle F \). | 1. Given | | 2. \( \overline{BO} \cong \overline{FO} \) | 2. Definition of midpoint | | 3. \( \angle B \cong \angle F \) | 3. Right Angle Congruence Theorem | | 4. \( \overline{AO} \cong \overline{EO} \) | 4. Given | | 5. \( \triangle ABO \cong \triangle EFO \) | 5. SAS | | 6. \( \angle A \cong \angle E \) | 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) | #### Explanation of Diagrams and Proofs: The proof involves showing that two right triangles \( \triangle ABO \) and \( \triangle EFO \) are congruent based on the SAS (Side-Angle-Side) criteria. To do this, several key statements and justifications are employed: 1. ***Point O is identified as the midpoint***, setting up the basis for why \( \overline{BO} \) is congruent to \( \overline{FO} \). 2. The fact that ***\( \angle B \) and \( \angle F \) are right angles*** confirms their congruence via the Right Angle Congruence Theorem. 3. ***\( \overline{AO} \) and \( \overline{EO} \) being congruent*** is given, as these are corresponding sides in the right
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