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Which statement is true about the diagram below? S R ARQC ASUC by SAS ARQC ASCU by SAS No triangles are congruent ARQC ASUC by SSS U

Which statement is true about the diagram below? S R ARQC ASUC by SAS ARQC ASCU by SAS No triangles are congruent ARQC ASUC by SSS U

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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AnswerPART 1 and PART 2

**Diagram Explanation:** The diagram shows two intersecting lines forming triangles \( \triangle RQC \) and \( \triangle SUC \). The following congruences are indicated by markings: - \( \overline{RQ} \) is marked as congruent to \( \overline{SU} \) with a single hash mark. - \( \overline{QC} \) is marked as congruent to \( \overline{CU} \) with a triple hash mark. - The angle \( \angle QCS \) is vertically opposite to \( \angle RCU \) and is therefore congruent, indicated by a common point C. **Question:** Which statement is true about the diagram below? - \( \triangle RQC \cong \triangle SUC \) by SAS - \( \triangle RQC \cong \triangle SCU \) by SAS - No triangles are congruent - \( \triangle RQC \cong \triangle SUC \) by SSS **Answer:** The correct answer is: - \( \triangle RQC \cong \triangle SUC \) by SAS Explanation: Using the SAS (Side-Angle-Side) congruence theorem, which states that two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, we can see that: - \( \overline{RQ} \cong \overline{SU} \) - \( \angle QCS \cong \angle SCU \) (vertically opposite angles) - \( \overline{QC} \cong \overline{CU} \) Therefore, \( \triangle RQC \cong \triangle SUC \) by SAS.
Transcribed Image Text:**Diagram Explanation:** The diagram shows two intersecting lines forming triangles \( \triangle RQC \) and \( \triangle SUC \). The following congruences are indicated by markings: - \( \overline{RQ} \) is marked as congruent to \( \overline{SU} \) with a single hash mark. - \( \overline{QC} \) is marked as congruent to \( \overline{CU} \) with a triple hash mark. - The angle \( \angle QCS \) is vertically opposite to \( \angle RCU \) and is therefore congruent, indicated by a common point C. **Question:** Which statement is true about the diagram below? - \( \triangle RQC \cong \triangle SUC \) by SAS - \( \triangle RQC \cong \triangle SCU \) by SAS - No triangles are congruent - \( \triangle RQC \cong \triangle SUC \) by SSS **Answer:** The correct answer is: - \( \triangle RQC \cong \triangle SUC \) by SAS Explanation: Using the SAS (Side-Angle-Side) congruence theorem, which states that two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, we can see that: - \( \overline{RQ} \cong \overline{SU} \) - \( \angle QCS \cong \angle SCU \) (vertically opposite angles) - \( \overline{QC} \cong \overline{CU} \) Therefore, \( \triangle RQC \cong \triangle SUC \) by SAS.
### Which statement, if true, would prove △ABC ≅ △RQP by SAS? #### Diagram Explanation The image showcases two triangles: △ABC and △RQP. Each triangle has markings indicating the length of specific sides and angles. - **For △ABC:** - Side AC is marked with two lines, indicating it is congruent to another side with the same marking. - Side AB is marked with one line, indicating it is congruent to another side with the same marking. - **For △RQP:** - Side RP is marked with two lines, indicating it is congruent to another side with the same marking. - Side PQ is marked with one line, indicating it is congruent to another side with the same marking. The triangles are positioned with vertex A corresponding to vertex R, vertex B to P, and vertex C to Q. #### Question and Options The question poses which statement would prove the triangles congruent by the Side-Angle-Side (SAS) theorem. Below are the options given: - ○ **BC ≅ PQ** - ○ **AB ≅ PQ** - ○ **∠P ≅ ∠C** - ○ **∠A ≅ ∠R** The goal is to select the correct statement that ensures the triangles are congruent by SAS, considering the markings provided.
Transcribed Image Text:### Which statement, if true, would prove △ABC ≅ △RQP by SAS? #### Diagram Explanation The image showcases two triangles: △ABC and △RQP. Each triangle has markings indicating the length of specific sides and angles. - **For △ABC:** - Side AC is marked with two lines, indicating it is congruent to another side with the same marking. - Side AB is marked with one line, indicating it is congruent to another side with the same marking. - **For △RQP:** - Side RP is marked with two lines, indicating it is congruent to another side with the same marking. - Side PQ is marked with one line, indicating it is congruent to another side with the same marking. The triangles are positioned with vertex A corresponding to vertex R, vertex B to P, and vertex C to Q. #### Question and Options The question poses which statement would prove the triangles congruent by the Side-Angle-Side (SAS) theorem. Below are the options given: - ○ **BC ≅ PQ** - ○ **AB ≅ PQ** - ○ **∠P ≅ ∠C** - ○ **∠A ≅ ∠R** The goal is to select the correct statement that ensures the triangles are congruent by SAS, considering the markings provided.
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