To use any of these reasons in a proof, you must have already stated - PowerPoint PPT Presentation

Review : If two parallel lines are cut by a transversal, then corresponding angles are congruent. If two parallel lines are cut by a transversal, then alternate interior angles are congruent. If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. If two parallel lines are cut by a transversal, then same side interior angles are supplementary. To use any of these reasons in a proof, you must have already stated that you have parallel lines.

Postulate If two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel. t 1 m 2 n

Theorem If two lines are cut by a transversal and the alternate interior/exterior angles are congruent, then the two lines are parallel. t 3 7 m 1 8 5 2 n 6 4

Theorem - Proof Given: Transversal t cuts m and n;  1 @  2 Prove: m //n 1.  1 @  2 1. Given 2. Vertical angles are congruent 2.  6 @  2 3. Substitution 3.  6 @  1 4. If two lines are cut by a transversal 4. m // n and corresponding angles are congruent then lines are parallel. 3 7 m 1 8 5 2 n 6 4 t

Theorem If two lines are cut by a transversal and the same side interior angles are supplementary, then the two lines are parallel. t 3 m 7 1 8 5 2 n 6 4

Theorem - Proof Given: Transversal t cuts m and n;  1 &  5 are supplementary. Prove: m //n 1.  1 &  5 are supplementary 1. Given 2. m  1 + m  5 = 180 2. Definition of supplementary angles 3. Angle Addition Postulate 3. m  5 + m  2 = 180 4. m  5 + m  2 = m  1 + m  5 4. Substitution 5. Subtraction 5. m  2 = m  1 6. If two lines are cut by a transversal and the 6. m // n alternate interior angles are congruent then the lines are parallel. 3 7 m 1 8 5 2 n 6 4 t

Theorem In a plane two lines perpendicular to the same line are parallel. n k t

Theorem - Proof 2 n Given: k ^ t; n ^ t Prove: k //n 5 k 1. k ^ t 1. Given t 2.  5 is a right angle. 2. Definition of perpendicular lines. 3. Definition of a right angle. 3. m  5 = 90 4. n ^ t 4. Given 5.  2 is a right angle. 5 . Definition of perpendicular lines. 6. Definition of a right angle. 6. m  2 = 90 7. Substitution 7. m  2 = m  5 8. If two lines are cut by a transversal and the corresponding 8. k // n angles are congruent, then the lines are parallel.

Theorem Through a point outside a line, there is exactly one line parallel to the given line.

Theorem Through a point outside a line, there is exactly one line perpendicular to the given line.

Theorem Two lines parallel to a third line are parallel to each other. If a // c and b // c, then a // b. a b c

Recap – 5 Ways to Prove Lines are Parallel 1. If two lines are cut by a transversal and CORRESPONDING ANGLES ARE CONGRUENT, then lines are parallel. 2. If two lines are cut by a transversal and ALTERNATE INTERIOR/EXTERIOR ANGLES ARE CONGRUENT, then the lines are parallel. 3. If two lines are cut by a transversal and SAME SIDE INTERIOR/EXTERIOR ANGLES ARE SUPPLEMENTARY then the lines are parallel. 4. If two lines are PERPENDICULAR TO A THIRD LINE, then the lines are parallel. 5. If two lines are PARALLEL TO A THIRD LINE, then the lines are parallel.

Practice: What two lines are parallel (if any) according to the given information? l m n 1. m  1 = m  4 l // m j 9 8 6 2. m  5 + m  6 = 180° j // k 1 2 3 4 5 k 3. m  8 = m  1 7 j // k 4. m  5 + m  4 = 180° None 5. m  1 = m  7 None j // k 6. m  8 + m  2 + m  3 = 180° l // m 9. m  6 + m  4 = 180 7. m  7 = m  4 None None 8. m  2 + m  3 = 180°