Geometry Definitions, Postulates, and Theorems Chapter : Parallel and Perpendicular Lines Section.1: Identify Pairs of Lines and Angles Standards: Prepare for 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. Parallel Lines Coplanar lines that do not intersect. Skew Lines Noncoplanar lines that do not intersect. Ceiling Floor Parallel Planes Two planes that do not intersect. Ceiling Floor Perpendicular Lines Lines that intersect to form 0 90 (right) angles. Segments and rays are parallel if they lie on parallel lines. Ex. AB ll CD in the diagram below. Ex. B A C D F E G H Think of each segment in the diagram as part of a line. a) Name a line parallel to BF. b) BC is skew to? AE & DH c) Name a line perpendicular to BC. d) Name a plane parallel to plane E F G. CG, DH, AE BF & CG BA & CD ABC Given this: Parallel Postulate IF there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. We can draw this: Perpendicular Postulate IF there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. (over)
t 1 5 6 7 8 Transversal A line that intersects two or more coplanar lines at different points. m n Corresponding Angles Angles in the same general location. Both are above the lines and to the left of the transversal, etc. Alternate Interior Angles Interior angles that are on opposite sides of the transversal. Alternate Exterior Angles Exterior angles that are on opposite sides of the transversal. Consecutive Interior Angles (same side interior angles) Interior angles that are on the same side of the transversal. Ex. a) Name an angle that corresponds to. b) Name a pair of alternate interior angles. c) Name a pair of alternate exterior angles. d) 6 and 7 are Consecutive Interior angles. 5 1 6 8 7
Section.: Use Parallel Lines and Transversals Standards: 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. 1 5 6 7 8 ***Corresponding Angles Postulate THEN the pairs of corresponding angles are congruent. The measures are equal. Triangles mark parallel lines ***Theorem.1 Alternate Interior Angles Theorem THEN the pairs of alternate interior angles are congruent. ***Theorem. Alternate Exterior Angles Theorem THEN the pairs of alternate exterior angles are congruent. ***Theorem. Consecutive Interior Angles Theorem THEN the pairs of consecutive interior angles are supplementary (add up to = 180). Find the missing angles. EX. EX. Find: 8 5 7 6 60 m m m m6 m7 m8 11 1 m1 m m m m5 (over)
EX. EX. x - 8 x x - 0 5 60 EX. x+ x-9 EX. Given: Prove: p II q m 1 m 180 0 1 5 6 p 7 8 q Statement Reason 1. p II q 1.. m1 m.... m + m = 180. 5. 5.
Section.: Prove Lines are Parallel Standards:.0 Students write geometric proofs, including proofs by contradiction. 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. In the last section: If two parallel lines are cut by a transversal, then the corresponding angles are congruent. In this section: If corresponding angles are congruent, then the lines are parallel. (converse) ***Corresponding Angles Converse Postulate IF two lines are cut by a transversal and the corresponding angles are congruent, Since corresponding angles are congruent, then the lines are parallel. ***Theorem. Alternate Interior Angles Converse IF two lines are cut by and transversal and the alternate interior angles are congruent, Since alternate interior angles are congruent, then the lines are parallel. ***Theorem.5 Alternate Exterior Angles Converse IF two lines are cut by and transversal and the alternate exterior angles are congruent, Since alternate exterior angles are congruent, then the lines are parallel. ***Theorem.6 Consecutive Interior Angles Converse IF two lines are cut by and transversal and the consecutive interior angles are supplementary (add up to 180), Since consecutive interior angles are supplementary, then the lines are parallel. ***Theorem.7 Transitive Property of Parallel Lines IF two lines are parallel to the same line, THEN they are parallel to each other. Ex. Can you prove lines c and d are parallel? Why? a) b) c) Yes, because alternate exterior angles are congruent. No, because consecutive interior angles aren't supposed to be congruent, they're supposed to be supplementary. (over) Yes, because corresponding angles are congruent.
Ex. Find the value of x that makes c II d. Ex. Find the value of x that makes c II d. c d x + 1 x - 5 x x Given: 5 6 6 A 5 B Prove: AD II BC Statement Reason D 6 C 1. 5 6 and 6 1.. 5... m Given: m p; m q Prove: p II q 1 p q Statement Reason 1. 1. Given. 1, are right angles... Definition of right angles.. Substitution 5. 1 5. 6. 6.