3.3 Parallel Lines & Transversals

Postulate 16 – Corresponding Angles Postulate If two  lines are cut by a transversal, then the pairs of corresponding s are . i.e. If l m, then 12. 1 2 l m

Section 3.3 Prove Lines are Parallel Theorems Theorem 3.4: Alternate Interior Angles Converse Theorem 3.5: Alternate Exterior Angles Converse Theorem 3.6: Consecutive Int. Angles Converse If alternate interior angles are congruent, then the lines are parallel. If alternate exterior angles are congruent, then the lines are parallel. If consecutive interior angles are supplementary, then the lines are parallel.

Apply the Corresponding Angles Converse EXAMPLE 1 Apply the Corresponding Angles Converse Find the value of x that makes m n. If corr s , then m n . Set angles = to each other. (3x + 5)o = 65o Corr. Angles Converse 3x = 60 Subtract 5 x = 20 Divide by 3

GUIDED PRACTICE for Example 1 Is there enough information in the diagram to conclude that m n? Explain. 75o Yes, the supplement to 105o is 75o , making corr. angles are congruent , so m n.

Proving lines parallel EXAMPLE 2 Proving lines parallel Given the information, state the postulate or theorem that proves lines a and b are parallel? a. 1  5 b. 2  5 c. m 2  m 4 = 180° d.  3  2 Is it possible to prove a // b if 1  4? Explain. Alt. Ext. Angles Converse Corr. Angles Converse Consec. Int. Angles Converse Alt. Int. Angles Converse No, angles would need to be supplementary

GUIDED PRACTICE for Example 2 Can you prove that lines a and b are parallel? Explain why or why not. Yes; Alt. Ext. Angles Converse.

GUIDED PRACTICE for Example 2 Can you prove that lines a and b are parallel? Explain why or why not. Yes; Corr. Angles Converse.

GUIDED PRACTICE for Example 2 Can you prove that lines a and b are parallel? Explain why or why not.

Prove the Alternate Interior Angles Converse EXAMPLE 3 Prove the Alternate Interior Angles Converse Prove that if alternate interior angles are congruent, then the lines are parallel. ∠4  ∠5 GIVEN : PROVE : g // h STATEMENTS REASONS 1. ∠4  ∠5 1. Given 2. ∠1  ∠4 2. Vertical ∠s Congruent 3. ∠1  ∠5 3. Transitive Prop. g // h 4. 4. Corr. Angles Converse

Section 3.3 Transitive Property of Parallel Lines Theorem Theorem 3.7: Transitive Property of Parallel Lines If 2 lines are parallel to the same line, then they are parallel to each other. If and , then .

Ex: Find: m1= m2= m3= m4= m5= m6= x= 125o 2 3 5 4 6 x+15o

Ex: Find: m1=55° m2=125° m3=55° m4=125° m5=55° m6=125° x=40° 125o 2 3 5 4 6 x+15o

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