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Kent notices that in parallelogram ABCD the consecutive (or adjacent) angles, 1 and 3,  sum to 180 degrees. This means that they are supplementary.  Will two adjacent angles of a parallelogram always add to 180 degrees? Why or why not? (Hint: What do you know about Sides AB and DC? If you let AD be a transversal, what does that imply about angles 1 and 2? What is the relationship between angles 2 and 3?)

Kent notices that in parallelogram ABCD the consecutive (or adjacent) angles, 1 and 3,  sum to 180 degrees. This means that they are supplementary.  Will two adjacent angles of a parallelogram always add to 180 degrees? Why or why not? (Hint: What do you know about Sides AB and DC? If you let AD be a transversal, what does that imply about angles 1 and 2? What is the relationship between angles 2 and 3?)

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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Kent notices that in parallelogram ABCD the consecutive (or adjacent) angles, 1 and 3,  sum to 180 degrees. This means that they are supplementary.  Will two adjacent angles of a parallelogram always add to 180 degrees? Why or why not? (Hint: What do you know about Sides AB and DC? If you let AD be a transversal, what does that imply about angles 1 and 2? What is the relationship between angles 2 and 3?) 

This diagram illustrates the concept of alternate angles formed by a transverse line cutting through two parallel lines. - **Lines:** - Two parallel lines are represented by the thick dashed blue lines. - A transverse line (line \( \overline{AB} \)) intersects the two parallel lines at points \( A \) and \( B \). - A second transverse line (line \( \overline{CD} \)) intersects these two parallel lines at points \( C \) and \( D \). - **Points:** - Four points of intersection are labeled as \( A \), \( B \), \( C \), and \( D \). - **Angles:** - Angle \( 1 \) is formed between line \( \overline{AB} \) and the lower parallel line at point \( A \). - Angle \( 2 \) is adjacent to angle \( 1 \), sharing the vertex at \( A \). - Angle \( 3 \) is formed between line \( \overline{CD} \) and the top parallel line at point \( D \). - Angle \( 4 \) is adjacent to angle \( 3 \), sharing the vertex at \( D \). - **Properties Involved:** - **Alternate Angles:** According to the alternate angle theorem, angle \( 1 \) is equal to angle \( 3 \), and angle \( 2 \) is equal to angle \( 4 \), as they lie on opposite sides of the transverse lines and between the parallel lines. - **Parallel Lines:** The properties of parallel lines ensure that these angle relationships hold true. Understanding such diagrams is crucial for comprehending the geometric principles surrounding parallel lines and angles.
Transcribed Image Text:This diagram illustrates the concept of alternate angles formed by a transverse line cutting through two parallel lines. - **Lines:** - Two parallel lines are represented by the thick dashed blue lines. - A transverse line (line \( \overline{AB} \)) intersects the two parallel lines at points \( A \) and \( B \). - A second transverse line (line \( \overline{CD} \)) intersects these two parallel lines at points \( C \) and \( D \). - **Points:** - Four points of intersection are labeled as \( A \), \( B \), \( C \), and \( D \). - **Angles:** - Angle \( 1 \) is formed between line \( \overline{AB} \) and the lower parallel line at point \( A \). - Angle \( 2 \) is adjacent to angle \( 1 \), sharing the vertex at \( A \). - Angle \( 3 \) is formed between line \( \overline{CD} \) and the top parallel line at point \( D \). - Angle \( 4 \) is adjacent to angle \( 3 \), sharing the vertex at \( D \). - **Properties Involved:** - **Alternate Angles:** According to the alternate angle theorem, angle \( 1 \) is equal to angle \( 3 \), and angle \( 2 \) is equal to angle \( 4 \), as they lie on opposite sides of the transverse lines and between the parallel lines. - **Parallel Lines:** The properties of parallel lines ensure that these angle relationships hold true. Understanding such diagrams is crucial for comprehending the geometric principles surrounding parallel lines and angles.
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