**Topic: Triangle Congruence Proof** **Given:** - \( AM \cong CP \) - \( CM \cong GP \) - \( C \) is the midpoint of \( AG \) **Prove:** - \( \triangle ACM \cong \triangle CGP \) **Diagram:** A geometric diagram depicts two triangles, \( \triangle ACM \) and \( \triangle CGP \), sharing a common line segment \( CM \) and arranged such that \( C \) is the midpoint of \( AG \). **Proof Structure:** | Statement | Reason | |----------------------------------|----------------------------| | 1. \( AM \cong CP \) | Given | | 2. \( CM \cong GP \) | Given | | 3. \( C \) is the midpoint of \( AG \) | Given | | 4. \( AC \cong CG \) | Definition of Midpoint | | 5. \( \triangle ACM \cong \triangle CGP \) | SAS | **Explanation of Diagram and Proof:** The diagram shows two triangles where: - \( C \) serves as the midpoint of line \( AG \), so segment \( AC \) is congruent to segment \( CG \) by definition. - The sides \( AM \) and \( CP \) are congruent as given. - Similarly, \( CM \) and \( GP \) are congruent as given. By applying the Side-Angle-Side (SAS) postulate, it is proved that \( \triangle ACM \) is congruent to \( \triangle CGP \). This postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. **Title: Analyzing Triangle Congruence Using Coordinate Geometry** **Instructions:** Use the coordinates below to find the length of each segment, then determine if \( \triangle ABC \cong \triangle DEF \). ### Given Coordinates: **\(\triangle ABC:\)** - \(A(2, -8)\) - \(B(-5, -2)\) - \(C(-7, 3)\) **\(\triangle DEF:\)** - \(D(-9, 7)\) - \(E(-11, 12)\) - \(F(-2, 1)\) ### Task: Calculate the following segment lengths: 1. \(AB =\) \_\_\_\_\_\_\_\_ 2. \(BC =\) \_\_\_\_\_\_\_\_ 3. \(CA =\) \_\_\_\_\_\_\_\_ 4. \(DE =\) \_\_\_\_\_\_\_\_ 5. \(EF =\) \_\_\_\_\_\_\_\_ 6. \(FD =\) \_\_\_\_\_\_\_\_ ### Question: - Are the two triangles congruent by SSS (Side-Side-Side)? **Blanks to fill:** - **Blank 1:** \_\_\_\_\_ - **Blank 2:** \_\_\_\_\_ - **Blank 3:** \_\_\_\_\_ - **Blank 4:** \_\_\_\_\_ - **Blank 5:** \_\_\_\_\_ - **Blank 6:** \_\_\_\_\_ - **Blank 7:** \_\_\_\_\_ **Explanation:** To determine if two triangles are congruent by the SSS postulate, calculate and compare their corresponding side lengths using the distance formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Complete the calculations for each side, then compare the sets of side lengths.

Topic: Triangle Congruence Proof Given: - \( AM \cong CP \) - \( CM \cong GP \) - \( C \) is the midpoint of \( AG \) Prove: - \( \triangle ACM \cong \triangle CGP \) Diagram: A geometric diagram depicts two triangles, \( \triangle ACM \) and \( \triangle CGP \), sharing a common line segment \( CM \) and arranged such that \( C \) is the midpoint of \( AG \). Proof Structure: | Statement | Reason | |----------------------------------|----------------------------| | 1. \( AM \cong CP \) | Given | | 2. \( CM \cong GP \) | Given | | 3. \( C \) is the midpoint of \( AG \) | Given | | 4. \( AC \cong CG \) | Definition of Midpoint | | 5. \( \triangle ACM \cong \triangle CGP \) | SAS | Explanation of Diagram and Proof: The diagram shows two triangles where: - \( C \) serves as the midpoint of line \( AG \), so segment \( AC \) is congruent to segment \( CG \) by definition. - The sides \( AM \) and \( CP \) are congruent as given. - Similarly, \( CM \) and \( GP \) are congruent as given. By applying the Side-Angle-Side (SAS) postulate, it is proved that \( \triangle ACM \) is congruent to \( \triangle CGP \). This postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Title: Analyzing Triangle Congruence Using Coordinate Geometry Instructions: Use the coordinates below to find the length of each segment, then determine if \( \triangle ABC \cong \triangle DEF \). ### Given Coordinates: \(\triangle ABC:\) - \(A(2, -8)\) - \(B(-5, -2)\) - \(C(-7, 3)\) \(\triangle DEF:\) - \(D(-9, 7)\) - \(E(-11, 12)\) - \(F(-2, 1)\) ### Task: Calculate the following segment lengths: 1. \(AB =\) \_\_\_\_\_\_\_\_ 2. \(BC =\) \_\_\_\_\_\_\_\_ 3. \(CA =\) \_\_\_\_\_\_\_\_ 4. \(DE =\) \_\_\_\_\_\_\_\_ 5. \(EF =\) \_\_\_\_\_\_\_\_ 6. \(FD =\) \_\_\_\_\_\_\_\_ ### Question: - Are the two triangles congruent by SSS (Side-Side-Side)? Blanks to fill: - Blank 1: \_\_\_\_\_ - Blank 2: \_\_\_\_\_ - Blank 3: \_\_\_\_\_ - Blank 4: \_\_\_\_\_ - Blank 5: \_\_\_\_\_ - Blank 6: \_\_\_\_\_ - Blank 7: \_\_\_\_\_ Explanation: To determine if two triangles are congruent by the SSS postulate, calculate and compare their corresponding side lengths using the distance formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Complete the calculations for each side, then compare the sets of side lengths.

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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