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3.1 Identify Pairs of Lines and Angles Objectives • Identify the relationships between two lines or two planes • Name angles formed by a pair of lines and a transversal Relationships Between Lines • Parallel Lines – lines that are coplanar and do not intersect; We use the symbol || to represent parallel lines, (i.e. AB || CD). • Skew Lines – lines that do not intersect and are not coplanar • Parallel Planes – two planes that do not intersect EXAMPLE 1 EXAMPLE 1 Identify Relationships in Space Think of each segment in the figure as part of a line. Which line(s) or plane(s) in the figure appear to fit the description? a. Line(s) parallel to CD and containing point A b. Line(s) skew to CD and containing point A c. Line(s) perpendicular to CD and containing point A d. Plane(s) parallel to plane EFG and containing point A EXAMPLE 1 (solution) Identify Relationships in Space a. AB , HG , and EF all appear parallel to CD , but only AB contains point A. b. Both AG and AH appear skew to CD and contain point A. c. BC, AD , DE , and FC all appear perpendicular to CD , but only AD contains point A. d. Plane ABC appears parallel to plane EFG and contains point A. EXAMPLE 2 EXAMPLE 2 Identify Parallel and Perpendicular Lines The given line markings show how the roads are related to one another. a. Name a pair of parallel lines. b. Name a pair of perpendicular lines. c. Is FE AC ? Explain. EXAMPLE 2 (solution) EXAMPLE 2 Identify Parallel and Perpendicular Lines a. MD FE b. MD c. FE is not parallel to AC , because MD is parallel to FE and by the Parallel Postulate there is exactly one line parallel to FE through M. BF YOUR TURN GUIDED PRACTICE 1. Look at the diagram in Example 1. Name the lines through point H that appear skew to CD. ANSWER AH , EH YOUR TURN GUIDED PRACTICE 2. In Example 2, can you use the Perpendicular Postulate to show that AC is not perpendicular to BF ? Explain why or why not. ANSWER Yes; Since A is not on MD and MD is to BF, the Perpendicular Postulate guarantees that there is exactly one line through a point perpendicular to a line, so AC can not be perpendicular to BF also. Postulates • Postulate 13 (Parallel Postulate) – There is exactly one line through a point that is parallel to a given line. • Postulate 14 (Perpendicular Postulate) – There is exactly one line through a point that is perpendicular to a given line. Relationships Between Lines and the Angles Formed • Transversal - a line that intersects two or more coplanar lines at different points 1 2 3 4 5 6 7 8 Angles 1 and 5 are corresponding angles Angles 1 and 8 are alternate exterior angles Relationships Between Lines and the Angles Formed 1 2 3 4 5 6 7 8 Angles 3 and 6 are alternate interior angles. Angles 3 and 5 are consecutive interior angles or same side interior angles EXAMPLE 3 EXAMPLE 3 Identify Angle Relationships Identify all pairs of angles of the given type. a. Corresponding b. Alternate interior c. Alternate exterior d. Consecutive interior SOLUTION a. 1 and 2 and 3 and 4 and 5 6 7 8 b. c. 2 and 7 4 and 5 1 and 8 3 and 6 d. 2 and 4 and 5 7 YOUR TURN GUIDED PRACTICE Classify the pair of numbered angles. 1. ANSWER corresponding angles YOUR TURN GUIDED PRACTICE Classify the pair of numbered angles. 2. ANSWER alternate exterior angles. Assignment • Geometry: Pg. 150 – 152 #3 – 33, 36