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Given: AB = Dc and AC = DB. Prove: AABCE ADCB. A B C AB DC gven AC DB g ven BC CB A ABC ADCB substitution, SSS O Reflexive Property of =, SAS Reflexive Pronerty of = SS.

Given: AB = Dc and AC = DB. Prove: AABCE ADCB. A B C AB DC gven AC DB g ven BC CB A ABC ADCB substitution, SSS O Reflexive Property of =, SAS Reflexive Pronerty of = SS.

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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The options A. Substitution, SSS B. Reflexive Property of = SAS C. Reflexive Property of = SSS D. Substitution, SAS
**Geometric Proof Example: Proving Triangles Congruent** **Given:** - \( AB \cong DC \) - \( AC \cong DB \) **Prove:** - \( \triangle ABC \cong \triangle DCB \) **Diagram Explanation:** The image shows a geometric figure with two triangles, \(\triangle ABC\) and \(\triangle DCB\), appearing to overlap as if placed in a kite shape. The vertices are labeled as \(A\), \(B\), \(C\), and \(D\). **Proof Steps:** 1. **Given Information:** - \( AB \cong DC \) (This is provided and needs no proof.) - \( AC \cong DB \) (This is also provided.) 2. **Conclusion:** - \( BC \cong CB \) (By the Reflexive Property, which states any geometric figure is congruent to itself.) 3. **Final Congruence Statement:** - \( \triangle ABC \cong \triangle DCB \) **Key Concepts:** - **Reflexive Property of Congruence:** Any segment or angle is always congruent to itself. - **Side-Side-Side (SSS) Congruence:** If all three sides of one triangle are congruent to all three sides of another triangle, the triangles are congruent. **Options for Completing the Proof:** - Substitution (SSS) - Reflexive Property of \( \equiv \) (SAS) - Reflexive Property of \( \equiv \) (SSS) In this proof, using the Reflexive Property and the SSS postulate validates that \(\triangle ABC\) is congruent to \(\triangle DCB\).
Transcribed Image Text:**Geometric Proof Example: Proving Triangles Congruent** **Given:** - \( AB \cong DC \) - \( AC \cong DB \) **Prove:** - \( \triangle ABC \cong \triangle DCB \) **Diagram Explanation:** The image shows a geometric figure with two triangles, \(\triangle ABC\) and \(\triangle DCB\), appearing to overlap as if placed in a kite shape. The vertices are labeled as \(A\), \(B\), \(C\), and \(D\). **Proof Steps:** 1. **Given Information:** - \( AB \cong DC \) (This is provided and needs no proof.) - \( AC \cong DB \) (This is also provided.) 2. **Conclusion:** - \( BC \cong CB \) (By the Reflexive Property, which states any geometric figure is congruent to itself.) 3. **Final Congruence Statement:** - \( \triangle ABC \cong \triangle DCB \) **Key Concepts:** - **Reflexive Property of Congruence:** Any segment or angle is always congruent to itself. - **Side-Side-Side (SSS) Congruence:** If all three sides of one triangle are congruent to all three sides of another triangle, the triangles are congruent. **Options for Completing the Proof:** - Substitution (SSS) - Reflexive Property of \( \equiv \) (SAS) - Reflexive Property of \( \equiv \) (SSS) In this proof, using the Reflexive Property and the SSS postulate validates that \(\triangle ABC\) is congruent to \(\triangle DCB\).
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