Section 3-1 Properties of Parallel Lines SPI 32E: solve problems involving complementary, supplementary, congruent, vertical or adjacent angles given measures expressed algebraically Objectives: Identify angles formed by two lines and a transversal Prove and use properties of parallel lines Recall Vocabulary Complementary – sum of angles equal 90º Supplementary – sum of angles equal 180º Vertical Angles – are congruent Adjacent Angles – angle that share a side and vertex

The Role of a Transversal line that intersects two coplanar lines at two distinct points. Pairs of eight angles have special names:

Properties of Parallel Lines Red arrows indicate parallel lines

Properties of Parallel Lines crosses PARALLEL lines! Theorems Apply ONLY when a transversal crosses PARALLEL lines! Name all the pairs of corresponding angles. Corresponding Angles Postulate 1  5 3  7 2  6 4  8 Are they congruent? Why? Name all the pairs of Alternate Interior Angles. Alternate Interior Angles Theorem 3  6 4  5 Are they congruent? Why? Name all the pairs of Same Side Interior Angles. Same side interior angles are supplementary 3  5 4  6 What is significant about same side interior angles?

Proof of Theorem 3-1: Alternate Interior Angles If a transversal intersects two parallel lines, then alternate interior angles are congruent. Given: a||b Prove: 1  3 a||b Given  1  3. 1  4 If lines are ||, then corresponding angles are congruent. 4  3 Vertical angles are congruent. 1  3 Transitive Property of Congruence.

Proof of Theorem 3-2: Same Side Interior Angles If 2 lines are parallel and cut by a transversal, then same-side interior angles are supplementary. Given: a||b Prove: 1 and 2 are supplementary Make a plan: Prove 1 + 2 = 180 Show 2 + 3 = 180 Show m 1 = m 3 Substitute a||b Given 2 +3 = 180 Angle Addition Postulate m1 = m3 Corresponding Angles are  m2 + m1 = 180 Substitute 1 and 2 are suppl. Def of supplementary angles

Real-world Connection This is an airport diagram of Pompano Beach Air Park in Florida. How can the following help construct an airport runway? Alternate angles Corresponding angles Same side interior angles

Finding Measures of Angles Find the measure of the angles by identifying which postulate or theorem justifies each answer. m1 m1 = 50 Corresponding Angles m2 m2 = 130 Same Side Interior m3 m3 = 130 Corresponding Angles m4 m4 = 130 Vertical Angles are  m5 m5 = 50 Vertical Angles are  m6 m6 = 50 Alt Interior Angles are  m7 m7 = 130 Same side interior angles m8 m8 = 50 Vertical angles are 

Using Algebra to Find Measures of Angles Find the values of x and y. What is the value of x? x = 70 Corr. Angles What is the value of y? 70 + 50 + y = 180 Angle Addition Postulate y = 60 Subtraction Prop. Of Eq

Using Algebra to Find Measures of Angles Find the values of x and y. Then find the measures of the angles. What relationship for x do you notice by looking at the diagram? 2x + 90 = 180 Same side interior angles = 180 x = 45 Sub and Div Property of Equality What is the relation between y and (y - 50)? y + (y - 50) = 180 Same side interior angles = 180 2y - 50 = 180 Simplify y = 115 Add and Div Property of Equality