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M Which method proves the triangles must be congruent?

M Which method proves the triangles must be congruent?

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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### Triangle Congruence #### Explanation and Diagram: The diagram above illustrates two triangles, △JKL and △NML, which share a common vertex, L, forming an X-like configuration. Key points in the diagram: - **Vertices**: The triangles have vertices labeled as J, K, L for the first triangle and N, M, L for the second triangle. - **Congruence indicators**: - **Angles**: ∠KJL and ∠MNL are marked to show that they are equal. - **Sides**: Sides JK and NM are marked, indicating they are of equal length. #### Question: Which method proves the triangles must be congruent? **Possible Methods to Consider**: - **SSS (Side-Side-Side)**: This method requires all three sides of one triangle to be congruent to all three sides of another triangle. - **SAS (Side-Angle-Side)**: This method requires two sides and the included angle of one triangle to be congruent to two sides and the included angle of another triangle. - **ASA (Angle-Side-Angle)**: This method requires two angles and the included side of one triangle to be congruent to two angles and the included side of another triangle. - **AAS (Angle-Angle-Side)**: This method requires two angles and a non-included side of one triangle to be congruent to the corresponding parts of another triangle. - **HL (Hypotenuse-Leg)**: This method is specific to right triangles and requires the hypotenuse and one leg to be congruent to the hypotenuse and one leg of another right triangle. **Analysis for This Diagram**: For triangles △JKL and △NML: - ∠KJL ≅ ∠MNL (as marked equal) - JL ≅ NL (common side) - JK ≅ NM (as marked) Using the **SAS (Side-Angle-Side)** congruence method: - Side JK ≅ Side NM (given/marked) - Angle ∠KJL ≅ Angle ∠MNL (given/marked) - Side JL ≅ Side NL (common to both triangles) Thus, by the SAS method, we can conclude that △JKL ≅ △NML. ##### Answer: The method that proves the triangles must be congruent is **SAS (Side-Angle
Transcribed Image Text:### Triangle Congruence #### Explanation and Diagram: The diagram above illustrates two triangles, △JKL and △NML, which share a common vertex, L, forming an X-like configuration. Key points in the diagram: - **Vertices**: The triangles have vertices labeled as J, K, L for the first triangle and N, M, L for the second triangle. - **Congruence indicators**: - **Angles**: ∠KJL and ∠MNL are marked to show that they are equal. - **Sides**: Sides JK and NM are marked, indicating they are of equal length. #### Question: Which method proves the triangles must be congruent? **Possible Methods to Consider**: - **SSS (Side-Side-Side)**: This method requires all three sides of one triangle to be congruent to all three sides of another triangle. - **SAS (Side-Angle-Side)**: This method requires two sides and the included angle of one triangle to be congruent to two sides and the included angle of another triangle. - **ASA (Angle-Side-Angle)**: This method requires two angles and the included side of one triangle to be congruent to two angles and the included side of another triangle. - **AAS (Angle-Angle-Side)**: This method requires two angles and a non-included side of one triangle to be congruent to the corresponding parts of another triangle. - **HL (Hypotenuse-Leg)**: This method is specific to right triangles and requires the hypotenuse and one leg to be congruent to the hypotenuse and one leg of another right triangle. **Analysis for This Diagram**: For triangles △JKL and △NML: - ∠KJL ≅ ∠MNL (as marked equal) - JL ≅ NL (common side) - JK ≅ NM (as marked) Using the **SAS (Side-Angle-Side)** congruence method: - Side JK ≅ Side NM (given/marked) - Angle ∠KJL ≅ Angle ∠MNL (given/marked) - Side JL ≅ Side NL (common to both triangles) Thus, by the SAS method, we can conclude that △JKL ≅ △NML. ##### Answer: The method that proves the triangles must be congruent is **SAS (Side-Angle
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