Section 2 1 The Parallel Postulate Special Angles

Section 2. 1 The Parallel Postulate & Special Angles 9/18/2020 1

Construction 6 Constructing a line perpendicular to a given line from a point not on the given line. (Figure 2. 1 p. 72) Steps: 1. From point P, using your compass, swipe an arc that intersects line l in two places. 2. Placing the compass point where the swipe intersects the line, open the compass past halfway and swipe below the line. Repeat step 2 from the other swipe to form an x With a straightedge connect point P and the x. Label the new line. • Theorem 2. 1. 1: From a point not on a given line, there is exactly one line perpendicular to the given line. 9/18/2020 2

Parallel Lines • Lines in the same plane that never intersect • Notation: l || m Line l Line m 9/18/2020 3

Several Applications of the Word Parallel Fig. 2. 3 p. 73 9/18/2020 4

It follows from definition of parallel lines that: • Portions (segments or rays) of parallel lines are parallel. Extensions of parallel segments or rays are parallel. • Postulate 10: (Parallel Postulate): Through a point not on a line, exactly one line is parallel to the given line. 9/18/2020 5

Transversal • A line that intersects 2 or more lines in different points. 1 4 3 6 5 7 8 9/18/2020 2 t 6

Transversal continued • At each point of intersection 4 angles are formed. • Angles that lie in the interior region – angles 3, 4, 5, & 6 – are interior angles. • Angles 1, 2, 7, & 8 lie in the exterior region are called exterior angles. 9/18/2020 7

Types of Angle Pairs Alternate interior angles (4 & 5, 3 & 6) • Angles are interior angles. • Angles are on opposite of the transversal. • Angles do not have the same vertex. • Hint: look for z oriented in any way. 9/18/2020 Corresponding angles (4 & 8, 3 & 7, 2 & 6, 1 & 5) • One angle is an interior angle; the other is an exterior angle. • Angles are on the same side of the transversal. • Angles do not have the same vertex. • Hint: Look for F, oriented any way. 8

|| lines & transversal • Corresponding angles are congruent. Ex. 1 p. 75 9/18/2020 9

|| lines & transversal • Alternate interior angles are congruent. 9/18/2020 10

|| lines & transversal • Interior angles on the same side of the transversal are supplementary. a b 9/18/2020 11

|| lines & transversal • Alternate exterior angles are congruent. 9/18/2020 12

Parallel Lines & Transversal • Postulate 11: If 2 parallel lines are cut by a transversal, then the corresponding angles are congruent. • Theorem 2. 1. 2: If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Proof p. 75 • Theorem 2. 1. 3: If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. • Theorem 2. 1. 4: If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary. Proof p. 76 • Theorem 2. 1. 5: If two parallel lines are cut by a transversal, then the exterior angles on the same side of the transversal are supplementary. Ex. 2, 3 p. 77 -8 9/18/2020 13