Parallel Lines & Transversals 3.3

Transversal A line, ray, or segment that intersects 2 or more COPLANAR lines, rays, or segments. Non-Parallel lines transversal Parallel lines transversal

1 5 Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent

Alternate Interior Angles Postulate If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Consecutive Interior Angles Postulate m 4 + m 5 = 180° m 3 + m 6 = 180° If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. m 4 + m 5 = 180° m 3 + m 6 = 180°

Alternate Exterior Angles Postulate If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 2 8

j k Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Prove the Alternate Interior Angles Theorem. GIVEN p || q StatementsReasons p || q 1 PROVE 1  2 2  1   3 3  3   2 4 1  21  2 1 Given 2 Corresponding Angles Postulate 3 Vertical Angles Theorem 4 Transitive property of Congruence

Using Properties of Parallel Lines Given that m  5 = 65°, find each measure. Tell which postulate or theorem you use. Linear Pair Postulate Alternate Exterior Angles Theorem Corresponding Angles Postulate Vertical Angles Theorem m  6 = m  5 = 65° m  7 = 180° – m  5 = 115° m 9 = m 7 = 115° m 8 = m 5 = 65°

m 4 = 125° m 4 + (x + 15)° = 180° Use properties of parallel lines to find the value of x. Corresponding Angles Postulate Linear Pair Postulate 125° + (x + 15)° = 180° Substitute. P ROPERTIES OF S PECIAL P AIRS OF A NGLES Subtract. x = 40°

Give an example of each angle pair. A. corresponding angles B. alternate interior angles C. alternate exterior angles 1 and 5 or 2 and 6 or 4 and 8 or 3 and 7 D. consecutive interior angles 3 and 5 or 4 and 6 1 and 7 or 2 and 8 3 and 6 or 4 and 5 GIVE AN EXAMPLE OF EACH ANGLE PAIR

A. corresponding angles B. alternate interior angles C. alternate exterior angles 1 and 3 D. consecutive interior angles 2 and 7 1 and 8 2 and 3 GIVE AN EXAMPLE OF EACH ANGLE PAIR

Special Angle Relationships Interior Angles  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 ♥Alternate Interior Angles are CONGRUENT ♥Alternate Exterior Angles are CONGRUENT ♥Consecutive Interior Angles are SUPPLEMENTARY ♥ Corresponding Angles are CONGRUENT ♥Consecutive Exterior Angles are SUPPLEMENTARY If the lines are not parallel, these angle relationships DO NOT EXIST.

Let’s Practice m  1=120° Find all the remaining angle measures ° 120°

Find the value of x, name the angles. a. b. c. d. e. f. g. h. i. x = 64 x = 75 x = 12 x = 40 x = 60 x = 90 x = 15 x = 20

How would you show that the given lines are parallel? a. a and b b. b and c c. d and f d. e and g e. a and c Corresponding `s Congruent Consecutive Interior `s Supplementary Corresponding `s Congruent Calculate the missing  Corresponding `s Congruent Consecutive Interior `s Supplementary 43

Find the value of each variable. 1. x2. y x = 2 y = 4

Find the value of x and y that make the lines parallel, name the angles. a. x b. y 2x + 2 = x + 56 x = 54 Corresponding `s Congruent y = 63 2(54) + 2 = 110 110 Consecutive Exterior `s are Supplementary y + 7 = 70 70 2(63) – 16 = 110 110

IDENTIFY THE TRANSVERSAL, & CLASSIFY EACH ANGLE PAIR p q r s a. 2 and 16 Alternate Exterior ’s Transversal p Lines r and s b. 6 and 7 Transversal r Lines p and q Consecutive Interior ’s

1 and 3 A. 1 and 3 2 and 6 B. 2 and 6 4 and 6 C. 4 and 6 transversal l corresponding s transversal n alternate interior s transversal m alternate exterior s IDENTIFY THE TRANSVERSAL, & CLASSIFY EACH ANGLE PAIR

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