Given: ABCD is a parallelogram, BEL AC and DFL AC. Prove: ABEC ADFA. m... rary data

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
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### Geometry Proof Exercise: Congruent Triangles

#### Given:
- \(ABCD\) is a parallelogram
- \(BE \perp AC\) and \(DF \perp AC\)

#### Prove:
- \(\triangle BEC \cong \triangle DFA\)

#### Proof Structure:
| Step | Statement                          | Reason               |
|------|------------------------------------|----------------------|
| 1    | \(ABCD\) is a parallelogram         | Given                |
|      | \(BE \perp AC\)                     | Given                |
|      | \(DF \perp AC\)                     | Given                |

#### Diagram Explanation:
The diagram features a parallelogram \(ABCD\). Points \(E\) and \(F\) are marked on \(AC\) such that \(BE \perp AC\) and \(DF \perp AC\). Lines \(BE\) and \(DF\) are perpendicular to \(AC\) at points \(E\) and \(F\) respectively. This sets up the context for proving the congruence of triangles \(BEC\) and \(DFA\).

Ensure to analyze the properties of the parallelogram, the relationships between the given angles and sides, and employ appropriate congruence postulates such as ASA (Angle-Side-Angle), SAS (Side-Angle-Side), or RHS (Right Hypotenuse Side) for right-angled triangles.

This exercise aims to reinforce understanding of geometric properties and the utilization of formal proof methods.
Transcribed Image Text:### Geometry Proof Exercise: Congruent Triangles #### Given: - \(ABCD\) is a parallelogram - \(BE \perp AC\) and \(DF \perp AC\) #### Prove: - \(\triangle BEC \cong \triangle DFA\) #### Proof Structure: | Step | Statement | Reason | |------|------------------------------------|----------------------| | 1 | \(ABCD\) is a parallelogram | Given | | | \(BE \perp AC\) | Given | | | \(DF \perp AC\) | Given | #### Diagram Explanation: The diagram features a parallelogram \(ABCD\). Points \(E\) and \(F\) are marked on \(AC\) such that \(BE \perp AC\) and \(DF \perp AC\). Lines \(BE\) and \(DF\) are perpendicular to \(AC\) at points \(E\) and \(F\) respectively. This sets up the context for proving the congruence of triangles \(BEC\) and \(DFA\). Ensure to analyze the properties of the parallelogram, the relationships between the given angles and sides, and employ appropriate congruence postulates such as ASA (Angle-Side-Angle), SAS (Side-Angle-Side), or RHS (Right Hypotenuse Side) for right-angled triangles. This exercise aims to reinforce understanding of geometric properties and the utilization of formal proof methods.
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