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
Related questions
Question
![**Problem 13:**
**Given:**
1. \( ED \parallel DB \), \( CF \parallel EF \)
2. \( AB \cong AC \)
**To Prove:**
\( \triangle BDE \cong \triangle CEF \)
And
\[ \frac{BD}{CF} = \frac{DE}{FE} \]
**Diagram:**
The diagram shows a large triangle \( \triangle ABC \) with \( D \) and \( E \) on side \( AB \), \( E \) and \( F \) on side \( AC \). Segments \( ED \) and \( CF \) are parallel to \( DB \) and \( EF \), respectively.
**Proof:**
| **Statements** | **Reasons** |
|----------------|-------------|
| 1. \( ED \parallel DB \), \( CF \parallel EF \) | Given |
| 2. \( AB \cong AC \) | Given |
| 3. \( \angle BDE \cong \angle CEF \) | Alternate Interior Angles (Parallel lines) |
| 4. \( \angle BED \cong \angle ECF \) | Alternate Interior Angles (Parallel lines) |
| 5. \( \triangle BDE \cong \triangle CEF \) | AA Similarity Criterion (Angle-Angle) |
| 6. \( \frac{BD}{CF} = \frac{DE}{FE} \) | Corresponding sides of similar triangles are proportional |
This structured approach ensures that students can follow the logical sequence of geometric proof while understanding the importance of each step and its reason.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ec0482a-ffdc-4040-bf9f-ac7e87cdc73c%2F41866a26-0a35-4df5-b2f1-acb81b3bcd6b%2Fu31i9tl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 13:**
**Given:**
1. \( ED \parallel DB \), \( CF \parallel EF \)
2. \( AB \cong AC \)
**To Prove:**
\( \triangle BDE \cong \triangle CEF \)
And
\[ \frac{BD}{CF} = \frac{DE}{FE} \]
**Diagram:**
The diagram shows a large triangle \( \triangle ABC \) with \( D \) and \( E \) on side \( AB \), \( E \) and \( F \) on side \( AC \). Segments \( ED \) and \( CF \) are parallel to \( DB \) and \( EF \), respectively.
**Proof:**
| **Statements** | **Reasons** |
|----------------|-------------|
| 1. \( ED \parallel DB \), \( CF \parallel EF \) | Given |
| 2. \( AB \cong AC \) | Given |
| 3. \( \angle BDE \cong \angle CEF \) | Alternate Interior Angles (Parallel lines) |
| 4. \( \angle BED \cong \angle ECF \) | Alternate Interior Angles (Parallel lines) |
| 5. \( \triangle BDE \cong \triangle CEF \) | AA Similarity Criterion (Angle-Angle) |
| 6. \( \frac{BD}{CF} = \frac{DE}{FE} \) | Corresponding sides of similar triangles are proportional |
This structured approach ensures that students can follow the logical sequence of geometric proof while understanding the importance of each step and its reason.
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