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Statements Reasons 1. a. 1. Given 2. /1= Z2, Z3= 4 2. b. 3. c. 3. Transitive Property 4. d. 4. Reflexive Property 5. e. 5. SAS

Statements Reasons 1. a. 1. Given 2. /1= Z2, Z3= 4 2. b. 3. c. 3. Transitive Property 4. d. 4. Reflexive Property 5. e. 5. SAS

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 image presents a geometric proof involving congruent angles and segments. Here's a detailed transcription and explanation suited for an educational website: --- ### Diagram and Proof Explanation Above is a diagram of two intersecting lines forming vertical angles at point "T". There are two triangles, \( \triangle MNT \) and \( \triangle TPM \). The given information and reasoning chain validate the congruency of these triangles using several geometric properties. ### Proof Flowchart: 1. **\( \angle 1 \cong \angle 4 \)** *Given.* 2. **\( \angle 1 \cong \angle 2 \); \( \angle 3 \cong \angle 4 \)** *Vertical angles are congruent.* 3. **\( \angle 2 \cong \angle 3 \)** *Transitive Property.* 4. **\( MN \cong PT \)** *Given.* 5. **\( MT \cong TM \)** *Reflexive Property.* 6. **\( \triangle MNT \cong \triangle TPM \)** *Side-Angle-Side (SAS) Congruence Postulate.* ### Proof Table: | Statements | Reasons | |--------------------------------------------|------------------------| | 1. Given information | 1. Given | | 2. \( \angle 1 \cong \angle 2, \angle 3 \cong \angle 4 \) | 2. Vertical angles are congruent | | 3. \( MN \cong PT \) | 3. Transitive Property | | 4. Reflexive Property | 4. Reflexive Property | | 5. SAS Postulate | 5. SAS | --- ### Graphical Explanation The flowchart breaks down the logical sequence used to establish triangle congruence. It highlights: - The use of the **Transitive Property** to establish angle congruence among the angles formed by intersecting lines. - Application of the **Reflexive Property** to show that segment MT is congruent to itself. - Conclusion by the **SAS Congruence Postulate**, confirming the two triangles are congruent. This instructional content helps students understand how specific geometric theorems and postulates are applied to establish triangle congruence.
Transcribed Image Text:The image presents a geometric proof involving congruent angles and segments. Here's a detailed transcription and explanation suited for an educational website: --- ### Diagram and Proof Explanation Above is a diagram of two intersecting lines forming vertical angles at point "T". There are two triangles, \( \triangle MNT \) and \( \triangle TPM \). The given information and reasoning chain validate the congruency of these triangles using several geometric properties. ### Proof Flowchart: 1. **\( \angle 1 \cong \angle 4 \)** *Given.* 2. **\( \angle 1 \cong \angle 2 \); \( \angle 3 \cong \angle 4 \)** *Vertical angles are congruent.* 3. **\( \angle 2 \cong \angle 3 \)** *Transitive Property.* 4. **\( MN \cong PT \)** *Given.* 5. **\( MT \cong TM \)** *Reflexive Property.* 6. **\( \triangle MNT \cong \triangle TPM \)** *Side-Angle-Side (SAS) Congruence Postulate.* ### Proof Table: | Statements | Reasons | |--------------------------------------------|------------------------| | 1. Given information | 1. Given | | 2. \( \angle 1 \cong \angle 2, \angle 3 \cong \angle 4 \) | 2. Vertical angles are congruent | | 3. \( MN \cong PT \) | 3. Transitive Property | | 4. Reflexive Property | 4. Reflexive Property | | 5. SAS Postulate | 5. SAS | --- ### Graphical Explanation The flowchart breaks down the logical sequence used to establish triangle congruence. It highlights: - The use of the **Transitive Property** to establish angle congruence among the angles formed by intersecting lines. - Application of the **Reflexive Property** to show that segment MT is congruent to itself. - Conclusion by the **SAS Congruence Postulate**, confirming the two triangles are congruent. This instructional content helps students understand how specific geometric theorems and postulates are applied to establish triangle congruence.
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