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Parallel Lines & Transversals 3.3 Transversal A line, ray, or segment that intersects 2 or more COPLANAR lines, rays, or segments. Parallel lines Non-Parallel lines transversal transversal If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Corresponding Angles Postulate 2 1 3 4 6 5 7 8 1 5 22 66 33 77 44 88 If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Alternate Interior Angles Postulate 2 1 3 4 6 5 7 8 3 5 4 6 If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. Consecutive Interior Angles Postulate 2 1 3 4 6 5 7 8 m 4 + m 5 = 180° m 3 + m 6 = 180° If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Alternate Exterior Angles Postulate 2 1 3 4 6 5 7 8 1 7 2 8 If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Perpendicular Transversal Theorem j k Prove the Alternate Interior Angles Theorem. GIVEN p || q PROVE 1 2 Statements 1 p || q Reasons 1 Given 2 1 3 2 Corresponding Angles Postulate 3 3 2 3 Vertical Angles Theorem 4 1 2 4 Transitive property of Congruence Using Properties of Parallel Lines Given that m5 = 65°, find each measure. Tell which postulate or theorem you use. m 6 = m 5 = 65° m 7 = 180° – m 5 = 115° Vertical Angles Theorem Linear Pair Postulate m 8 = m 5 = 65° Corresponding Angles Postulate m 9 = m 7 = 115° Alternate Exterior Angles Theorem PROPERTIES OF SPECIAL PAIRS OF ANGLES Use properties of parallel lines to find the value of x. m m 4 = 125° 4 + (x + 15)° = 180° 125° + (x + 15)° = 180° x = 40° Corresponding Angles Postulate Linear Pair Postulate Substitute. Subtract. GIVE AN EXAMPLE OF EACH ANGLE PAIR Give an example of each angle pair. A. corresponding angles 1 and 5 or 2 and 6 or 4 and 8 or 3 and 7 B. alternate interior angles 3 and 5 or 4 and 6 C. alternate exterior angles 1 and 7 or 2 and 8 D. consecutive interior angles 3 and 6 or 4 and 5 GIVE AN EXAMPLE OF EACH ANGLE PAIR A. corresponding angles 1 and 3 B. alternate interior angles 2 and 7 C. alternate exterior angles 1 and 8 D. consecutive interior angles 2 and 3 Special Angle Relationships Interior Angles 1 3 4 5 6 7 8 2 3 & 6 are Alternate Interior angles 4 & 5 are Alternate Interior angles 3 & 5 are Consecutive Interior angles 4 & 6 are Consecutive Interior angles Exterior Angles 1 & 8 are Alternate Exterior angles 2 & 7 are Alternate Exterior angles 1 & 7 are Consecutive Exterior angles 2 & 8 are Consecutive Exterior angles Special Angle Relationships WHEN THE LINES ARE PARALLEL 1 3 5 2 ♥Alternate Interior Angles are CONGRUENT 4 6 7 8 ♥Alternate Exterior Angles are CONGRUENT ♥Consecutive Interior Angles are SUPPLEMENTARY If the lines are not parallel, these angle relationships DO NOT EXIST. ♥ Corresponding Angles are CONGRUENT ♥Consecutive Exterior Angles are SUPPLEMENTARY Let’s Practice 120°1 60°3 120° 5 60° 7 2 60° 4 120° 6 60° 8 120° m1=120° Find all the remaining angle measures. Find the value of x, name the angles. a. x = 64 b. x = 75 c. x = 12 d. x = 40 e. x = 60 f. x = 60 g. x = 90 h. x = 15 i. x = 20 How would you show that the given lines are parallel? 43 a. a and b Consecutive Interior `s Supplementary d. e and g b. b and c Corresponding `s Congruent e. a and c Calculate the missing Consecutive Interior Corresponding `s Supplementary `s Congruent c. d and f Corresponding `s Congruent Find the value of each variable. 1. x x = 2 2. y y = 4 Find the value of x and y that make the lines parallel, 110 name the angles. 110 70 a. x 110 b. y 2x + 2 = x + 56 y + 7 = 70 x = 54 y = 63 2(54) + 2 = 110 2(63) – 16 = 110 Corresponding `s Consecutive Exterior Congruent `s are Supplementary IDENTIFY THE TRANSVERSAL, & CLASSIFY EACH ANGLE PAIR a. 2 and 16 q p 1 2 8 7 3 4 6 5 Transversal p Lines r and s r Alternate Exterior ’s b. 6 and 7 Transversal r Lines p and q 9 10 11 12 16 15 14 13 Consecutive Interior ’s s IDENTIFY THE TRANSVERSAL, & CLASSIFY EACH ANGLE PAIR A. 1 and 3 transversal l corresponding s B. 2 and 6 transversal n alternate interior s C. 4 and 6 transversal m alternate exterior s Review If two lines are intersected by a transversal and any of the angle pairs shown below are congruent, then the lines are parallel. This fact is used in the construction of parallel lines. Assignment 3.3A and 3.3B Section 9 - 33