Properties of Parallel Lines: Theorems and Postulates Explained
Explore the fundamental theorems and postulates related to parallel lines, including Same-Side Interior Angles Postulate, Alternate Interior Angles Theorem, Corresponding Angles Theorem, and Alternate Exterior Angles Theorem. Learn how these geometric principles can help solve angle measurements and proofs.
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Presentation Transcript
Section 3-2 Properties of Parallel Lines, Calculations. Michael Schuetz
Postulate 3-1: Same-Side Interior Angles Postulate. Postulate If l//m parallel lines, then same- side interior angles are supplementary l l m m 1 2 l//m 4 3 If a transversal intersects two 5 6 8 7 + + m m = = 4 5 1 0 8 m Then 3 6 1 0 8 m
Theorem 3-1: Alternate Interior Angles Theorem. (AIA) Theorem If l//m parallel lines, then alternate interior angles are congruent. l l m m 1 2 l//m 4 3 If a transversal intersects two 5 6 8 7 4 6 Then 3 5
Theorem 3-2: Corresponding Angles Theorem. (Corr ) Theorem If l//m parallel lines, then corresponding angles are congruent. 's l l m m 1 2 l//m 4 3 If a transversal intersects two 5 6 8 7 1 5 3 7 Then 4 8 2 6
Theorem 3-3: Alternate Exterior Angles Theorem. (AEA) Theorem If l//m parallel lines, then Alternate exterior angles are congruent. l l m m 1 2 l//m 4 3 If a transversal intersects two 5 6 8 7 1 7 Then 2 8
Example 1, Angle Measure What are the measures of angles 3 and 4? Which theorem or postulate justifies your answer? m = 105o 3 p p q q AIA Theorem l l m m 2 5 m = Same-Side Interior Postulate 75o 4 8 1 3 105 4 6 7
Example 2, Angle Measure What is the value of y? Which theorem or postulate justifies your answer? Same-Side Interior Postulate tells us that p p q q l l ( ( ) ) + + = o o o o 40 80 180 y y 80 + = 40 o o o 40 1 00 y y = o o 60
Classwork: P. 153, # s 7, 12, 15, 16, 19, 20
Section 3-2 Properties of Parallel Lines, Proofs.
Proof of Theorem 3-1: AIA Given: l//m Prove: 4 6 l l m m 3 4 6 Statements Reasons 1. / / l 2. m 3. m 4. m 5. m 6. 4 m 3 3 3 4 1. Given 2. Supplementary Angles 3. Same-Side Interior Angles Postulate 4. Transitive Property of Equality + + + = = = = 180o 180o m + 4 6 4 6 m m m m 6 3 6 m 5. Subtraction Property of Equality 6. Definition of Congruence
Proving an angle relationship Given: l//m Prove: are supplementary m m l l 1 1 and 8 5 8 7 Reasons Statements 1. / / l 2. 1 3. m = 4. 5 and 8 are Suppl. 5. 5 m m + = 6. 1 m m + = 7. 1 and 8 are Suppl. m 1. Given 2. If lines are //, then Corr 3. have equal measure. 's 4. that form a linear pair are suppl. 's 's 5 m ' are s 1 5 8 8 180 180 5. Def. of suppl. 6. Substitution Property 's 7. Def. of suppl.
Homework: P. 153, # s 10, 11, 25, 26.
