Objective: To indentify angles formed by two lines and a transversal. Chapter 3 Lesson 1 Objective: To indentify angles formed by two lines and a transversal.

transversal: a line that intersects two coplanar lines at two distinct points. m transversal alternate interior angles: nonadjacent interior angles that lie on opposite sides of the transversal. t l 1 4 m 3 2

same-side interior angles: lie on the same side of the transversal and between l and m. 1 4 m 3 2 corresponding angles: lie on the same side of the transversal and in corresponding positions relative to l and m. t l 6 5 1 4 m 3 2 8 7

Example 1: Identifying Angles Name a pair of alternate interior angles and a pair of same-side interior angles. t l 6 5 3 and 4 are alternate interior angles 1 and 3 are same-side interior angles. 4 1 m 2 3 7 8

Example 2: Identifying Angles Name all pairs of corresponding angles. t l 1 2 3 4 5 6 7 8 6 5 5 and 2; 4 and 7 ; 6 and 3; 1 and 8 4 1 m 2 3 7 8

Example 3: Identifying Angles Identify which angle forms a pair of same-side interior angles with 1. Identify which angle forms a pair of corresponding angles with 1. Same-Side 8 4 1 8 5 3 2 7 6 Corresponding 5

Postulate 3-1: Corresponding Angles Postulate If a transversal intersects two parallel lines, then corresponding angles are congruent. t 1 ► l 2 ► m Theorem 3-1: Alternate Interior Angles Theorem If a transversal intersects two parallel lines, then alternate interior angles are congruent. t ► l 1 2 ► m

Theorem 3-2: Same-Side Interior Angles Theorem If a transversal intersects two parallel lines, then same-side interior angles are supplementary. t ► l 1 2 ► m

Two-Column Proof of Theorem 3-1 two-column proof: steps that prove a theorem Example 4 : Two-Column Proof of Theorem 3-1 t Given: 4 ► a 3 1 Prove: ► b Statements Reasons 1.) 2.) 3.) 4.) Given If lines are , then corr. are Vertical angles are congruent Transitive Property of Congruence

Two-Column Proof of Theorem 3-2 Example 5: Two-Column Proof of Theorem 3-2 t Given: 3 ► a 2 Prove: are supplementary. 1 ► b Statements Reasons 1.) 2.) 3.) 4.) 5.) Given Angle Add. Postulate Corresponding Angles Postulate Substitute Def. of Supplementary Angles are supplementary

Example 6: Finding Measures of Angles ◄ ▲ a b c d 1 5 3 4 2 6 7 8 50° Find ,and then . Which theorem or postulate justifies each answer? Since , because corresponding angles are congruent (Corresponding Angles Postulate). Since , because same-side interior angles are supplementary (Same-Side Interior Angles Theorem).

Example 7: Finding Measures of Angles Find the measure of each angle. Justify each answer. ◄ ▲ a b c d 1 5 3 4 2 6 7 8 50° a.) 130; corr. angles are congruent b.) 130; vert. angles are congruent c.) 50; alt. int. angles are congruent d.) 50; alt. int. angles are congruent e.) 130; same-side int. angles are supp. f.) 50; corr. angles are congruent or vert. angles are congruent

Example 8: Using Algebra to Find Angle Measures Find the values of x and y. x=70 70+50+y=180 120+y=180 y=60 Corresponding angles ▲ ▲ 50° Angle Add. Post. x° y° 70° Simplify Subtr. Prop of Equality

Example 9: Using Algebra to Find Angle Measures ► a° c° b° Find the values of a, b, and c. m 65° 40° ► a = 65 Alt. Int. Angles c=40 Alt. Int. Angles 65+40+b = 180 Angle Add. Post. 105+b=180 Simplify b=75 Subtr. Prop. Of Equality

Assignment pg.118-120 #1-25