3.3 Parallel Lines & Transversals
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 j k
Example 1: Proving the Alternate Interior Angles Theorem Given: p ║ q Prove: 1 ≅ 2 1 2 3
Proof Statements: p ║ q 1 ≅ 3 3 ≅ 2 1 ≅ 2 Reasons: Given Corresponding Angles Postulate Vertical Angles Theorem 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 8 5 7
Solutions: m 6 = m 5 = 65° m 7 = 180° - m 5 =115° Vertical Angles Theorem m 7 = 180° - m 5 =115° Linear Pair postulate m 8 = m 5 = 65° Corresponding Angles Postulate 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? k j 1 120°
Solution m 1 + 120° = 180° m 1 = 60° Consecutive Interior angles Theorem 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: m4 = 125° m4 +(x+15)°=180° 125°+(x+15)°= 180° Reasons: Corresponding Angles Postulate Linear Pair Postulate Substitution POE 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!!!!