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7. 10 8 9 5 D 3 E F G AC DG and B and F are midpoints of AC and DG, Subtraction Property

7. 10 8 9 5 D 3 E F G AC DG and B and F are midpoints of AC and DG, Subtraction Property

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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### Transcription and Explanation #### Diagram Description The image contains a geometric diagram with a triangle \( \triangle ABC \) and a line \( DEFG \). Point \( B \) is on \( AC \), and points \( D, E, F, \) and \( G \) are on the straight line. The segment \( AC \) is congruent to the segment \( DG \). Points \( B \) and \( F \) are midpoints of \( AC \) and \( DG \), respectively. #### Problem and Logical Steps: 1. **Given:** - \( AC \cong DG \) - \( B \) and \( F \) are midpoints of \( AC \) and \( DG \). **Question: Why is \( AB \cong DF \)?** - **Property Used:** Subtraction Property 2. **If \( \angle ABF \cong \angle FBC \) and \( \angle 8 \cong \angle 9 \), why is \( \angle 7 \cong \angle 10 \)?** - **Property Used:** Addition Property 3. **If \( \angle 7 \cong \angle 8 \) and \( \angle 8 \cong \angle 9 \), why is \( \angle 7 \cong \angle 9 \)?** - **Property Used:** Transitive Property 4. **If \( \angle 1 \cong \angle 6 \), why is \( \angle 2 \cong \angle 5 \)?** - **Property Used:** Division Property 5. **If \( \angle 7 \cong \angle 10 \), why is \( \angle ABG \cong \angle EBC \)?** - **Property Used:** Congruent Supplements Theorem This sequence of logical steps helps demonstrate various geometric properties used in proving congruence and equality in angles and segments. Each property, such as the Subtraction, Addition, Transitive, and Division Properties, as well as the Congruent Supplements Theorem, plays a crucial role in understanding geometric relationships.
Transcribed Image Text:### Transcription and Explanation #### Diagram Description The image contains a geometric diagram with a triangle \( \triangle ABC \) and a line \( DEFG \). Point \( B \) is on \( AC \), and points \( D, E, F, \) and \( G \) are on the straight line. The segment \( AC \) is congruent to the segment \( DG \). Points \( B \) and \( F \) are midpoints of \( AC \) and \( DG \), respectively. #### Problem and Logical Steps: 1. **Given:** - \( AC \cong DG \) - \( B \) and \( F \) are midpoints of \( AC \) and \( DG \). **Question: Why is \( AB \cong DF \)?** - **Property Used:** Subtraction Property 2. **If \( \angle ABF \cong \angle FBC \) and \( \angle 8 \cong \angle 9 \), why is \( \angle 7 \cong \angle 10 \)?** - **Property Used:** Addition Property 3. **If \( \angle 7 \cong \angle 8 \) and \( \angle 8 \cong \angle 9 \), why is \( \angle 7 \cong \angle 9 \)?** - **Property Used:** Transitive Property 4. **If \( \angle 1 \cong \angle 6 \), why is \( \angle 2 \cong \angle 5 \)?** - **Property Used:** Division Property 5. **If \( \angle 7 \cong \angle 10 \), why is \( \angle ABG \cong \angle EBC \)?** - **Property Used:** Congruent Supplements Theorem This sequence of logical steps helps demonstrate various geometric properties used in proving congruence and equality in angles and segments. Each property, such as the Subtraction, Addition, Transitive, and Division Properties, as well as the Congruent Supplements Theorem, plays a crucial role in understanding geometric relationships.
**Given:** \( \overline{TM} \) bisects \(\angle OTX\) **Prove:** \(\angle 1 \cong \angle 2\) ### Diagram Explanation In the diagram, lines intersect at point \( T \), forming several angles. The lines labeled \( PR \) and \( JM \) cross at point \( T \), forming angles \( \angle 1 \), \( \angle 2 \), \( \angle 3 \), and \( \angle 4 \). Additionally, line \( \overline{TM} \) bisects \(\angle OTX\), which means it divides \(\angle OTX\) into two equal parts. ### Proof Structure #### Statement and Reason 1. **Statement:** \( \overline{TM} \) bisects \(\angle OTX\) - **Reason:** Given 2. **Statement:** \(\angle 1 \cong \angle 3\) - **Reason:** Definition of vertical angles 3. **Statement:** \(\angle 2 \cong \angle 4\) - **Reason:** Definition of vertical angles 4. **Statement:** \(\angle 1 \cong \angle 2\) - **Reason:** Transitive property ### Options to Complete the Proof - **\(\angle 1 \cong \angle 2\)** - **\(\angle 1 \cong \angle 3\)** - **\(\angle 2 \cong \angle 4\)** - **Definition of vertical angles** - **Transitive property** - **\( \overline{TM} \) bisects \(\angle OTX\)** This structured approach helps you understand and prove that angles \(1\) and \(2\) are congruent using vertical angles and properties of angle bisectors.
Transcribed Image Text:**Given:** \( \overline{TM} \) bisects \(\angle OTX\) **Prove:** \(\angle 1 \cong \angle 2\) ### Diagram Explanation In the diagram, lines intersect at point \( T \), forming several angles. The lines labeled \( PR \) and \( JM \) cross at point \( T \), forming angles \( \angle 1 \), \( \angle 2 \), \( \angle 3 \), and \( \angle 4 \). Additionally, line \( \overline{TM} \) bisects \(\angle OTX\), which means it divides \(\angle OTX\) into two equal parts. ### Proof Structure #### Statement and Reason 1. **Statement:** \( \overline{TM} \) bisects \(\angle OTX\) - **Reason:** Given 2. **Statement:** \(\angle 1 \cong \angle 3\) - **Reason:** Definition of vertical angles 3. **Statement:** \(\angle 2 \cong \angle 4\) - **Reason:** Definition of vertical angles 4. **Statement:** \(\angle 1 \cong \angle 2\) - **Reason:** Transitive property ### Options to Complete the Proof - **\(\angle 1 \cong \angle 2\)** - **\(\angle 1 \cong \angle 3\)** - **\(\angle 2 \cong \angle 4\)** - **Definition of vertical angles** - **Transitive property** - **\( \overline{TM} \) bisects \(\angle OTX\)** This structured approach helps you understand and prove that angles \(1\) and \(2\) are congruent using vertical angles and properties of angle bisectors.
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