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PROOF CONCLUSIONS (1) Parallel lines and m intersect a transversal t at P and Q, with alternate interior angles 21 and 22. (2) Suppose m/1 # m2, and, for sake of argument, m/1 > m/2. (The proof is virtually the same if m21

PROOF CONCLUSIONS (1) Parallel lines and m intersect a transversal t at P and Q, with alternate interior angles 21 and 22. (2) Suppose m/1 # m2, and, for sake of argument, m/1 > m/2. (The proof is virtually the same if m21

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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what would be the justifications for 4,5, and 6

**Proof of Conditional Statement #2 (Theorem 2)** **PROOF** | CONCLUSIONS | JUSTIFICATIONS | |----------------------------------------------------------------------------|-------------------------------------------| | (1) Parallel lines \( \ell \) and \( m \) intersect a transversal \( t \) at \( P \) and \( Q \), with alternate interior angles \( \angle 1 \) and \( \angle 2 \). | Given | | (2) Suppose \( m \angle 1 \neq m \angle 2 \), and, for sake of argument, \( m \angle 1 > m \angle 2 \). (The proof is virtually the same if \( m \angle 1 < m \angle 2 \).) | Assumption for indirect proof | | (3) Construct ray \( \overrightarrow{PR} \) on the opposite side of \( t \) as \( \angle 2 \) so that \( m \angle QPR = m \angle 2 \). | Angle Construction Theorem | | (4) Then line \( \overline{RP} \) (\( = m' \)) is parallel to \( \ell \). | _________? | | (5) Thus, \( m' \parallel \ell \) and \( m \parallel \ell \). | _________? | | (6) Therefore, \( m \angle 1 = m \angle 2 \) or \( \angle 1 \cong \angle 2 \). | _________? | **Diagrams** The proof includes a diagram with two parts: 1. **Left Diagram**: Shows parallel lines \( \ell \) and \( m \) with a transversal \( t \) intersecting them at points \( P \) and \( Q \). Alternate interior angles \( \angle 1 \) and \( \angle 2 \) are marked at points \( P \) and \( Q \), respectively. 2. **Right Diagram**: Illustrates the constructed ray \( \overrightarrow{PR} \) on the opposite side of \( t \) from \( \angle 2 \), forming angle \( \angle QPR = \angle 2 \). The new line \( m' \), which is shown as parallel to \( \ell \), is also depicted.
Transcribed Image Text:**Proof of Conditional Statement #2 (Theorem 2)** **PROOF** | CONCLUSIONS | JUSTIFICATIONS | |----------------------------------------------------------------------------|-------------------------------------------| | (1) Parallel lines \( \ell \) and \( m \) intersect a transversal \( t \) at \( P \) and \( Q \), with alternate interior angles \( \angle 1 \) and \( \angle 2 \). | Given | | (2) Suppose \( m \angle 1 \neq m \angle 2 \), and, for sake of argument, \( m \angle 1 > m \angle 2 \). (The proof is virtually the same if \( m \angle 1 < m \angle 2 \).) | Assumption for indirect proof | | (3) Construct ray \( \overrightarrow{PR} \) on the opposite side of \( t \) as \( \angle 2 \) so that \( m \angle QPR = m \angle 2 \). | Angle Construction Theorem | | (4) Then line \( \overline{RP} \) (\( = m' \)) is parallel to \( \ell \). | _________? | | (5) Thus, \( m' \parallel \ell \) and \( m \parallel \ell \). | _________? | | (6) Therefore, \( m \angle 1 = m \angle 2 \) or \( \angle 1 \cong \angle 2 \). | _________? | **Diagrams** The proof includes a diagram with two parts: 1. **Left Diagram**: Shows parallel lines \( \ell \) and \( m \) with a transversal \( t \) intersecting them at points \( P \) and \( Q \). Alternate interior angles \( \angle 1 \) and \( \angle 2 \) are marked at points \( P \) and \( Q \), respectively. 2. **Right Diagram**: Illustrates the constructed ray \( \overrightarrow{PR} \) on the opposite side of \( t \) from \( \angle 2 \), forming angle \( \angle QPR = \angle 2 \). The new line \( m' \), which is shown as parallel to \( \ell \), is also depicted.
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