The Parallel Postulate & Special Angles Section 2.1 The Parallel Postulate & Special Angles 2/21/2019

Construction 6 Constructing a line perpendicular to a given line from a point not on the given line. (Figure 2.1 p. 72) Steps: From point P, using your compass, swipe an arc that intersects line l in two places. 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. 2/21/2019

Parallel Lines Lines in the same plane that never intersect Notation: l || m Line l Line m 2/21/2019

Several Applications of the Word Parallel Fig. 2.3 p. 73 2/21/2019

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. 2/21/2019

Transversal A line that intersects 2 or more lines in different points. 1 2 4 3 6 5 7 8 t 2/21/2019

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. 2/21/2019

Types of Angle Pairs Alternate interior angles Corresponding 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. 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. 2/21/2019

|| lines & transversal Corresponding angles are congruent. Ex. 1 p. 75 2/21/2019

|| lines & transversal Alternate interior angles are congruent. 2/21/2019

|| lines & transversal Interior angles on the same side of the transversal are supplementary. a b 2/21/2019

|| lines & transversal Alternate exterior angles are congruent. 2/21/2019

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 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 2/21/2019