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

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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

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

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

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