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3.3 Parallel Lines & Transversals Ms. Reser Standards/Objectives: Standard 3: Students will learn and apply geometric concepts. Objectives: • Prove and use results about parallel lines and transversals. • Use properties of parallel lines to solve real-life problems, such as estimating the Earth’s circumference Homework • Pgs. 146-148 #1-30 • REMINDER: There is a quiz after 3.3. If you want a glance at what it kind of looks like, check out pg. 149. You will be doing this for homework next class meeting. Postulate 15 Corresponding Angles Postulate • If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2 1 ≅ 2 Theorem 3.4 Alternate Interior Angles • If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3 4 3 ≅ 4 Theorem 3.5 Consecutive Interior Angles • If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. 5 6 5 + 6 = 180° Theorem 3.6 Alternate Exterior Angles • If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 7 8 7 ≅ 8 Theorem 3.7 Perpendicular Transversal • If a transversal is perpendicular to one of the two parallel lines, then it is perpendicular to the other. j h k jk Example 1: Proving the Alternate Interior Angles Theorem • Given: p ║ q • Prove: 1 ≅ 2 1 2 3 Proof Statements: 1. p ║ q 2. 1 ≅ 3 3. 3 ≅ 2 4. 1 ≅ 2 Reasons: 1. Given 2. Corresponding Angles Postulate 3. Vertical Angles Theorem 4. Transitive Property of Congruence Example 2: Using properties of parallel lines • Given that m 5 = 65°, find each measure. Tell which postulate or theorem you use. • A. m 6 B. m 7 • C. m 8 D. m 9 9 6 5 7 8 Solutions: a. m 6 = m 5 = 65° • Vertical Angles Theorem b. m 7 = 180° - m 5 =115° • Linear Pair postulate c. m 8 = m 5 = 65° • Corresponding Angles Postulate d. m 9 = m 7 = 115° • Alternate Exterior Angles Theorem Ex. 3—Classifying Leaves BOTANY—Some plants are classified by the arrangement of the veins in their leaves. In the diagram below, j ║ k. What is m 1? j k 120° 1 Solution 1. m 1 + 120° = 180° 2. m 1 = 60° 1. Consecutive Interior angles Theorem 2. Subtraction POE Ex. 4: Using properties of parallel lines • Use the properties of parallel lines to find the value of x. 125° 4 (x + 15)° Proof Statements: 1. m4 = 125° 2. m4 +(x+15)°=180° 3. 125°+(x+15)°= 180° 4. x = 40° Reasons: 1. Corresponding Angles Postulate 2. Linear Pair Postulate 3. Substitution POE 4. Subtraction POE NOTE: • You must show all your work. Check your syllabus . . . it tells you everything I expect. We are moving into the next quarter shortly, and I expect that your work will be even more professional, neat, organized, and will show even at a casual glance that you did your homework. IF IT EVEN LOOKS COPIED . . . NO CREDIT!!!!