Parallel and Perpendicular Lines Chapter 3

Objectives: Identify relationships between the angles formed by two lines and a transversal, especially when the two lines are parallel. Use properties of parallel and perpendicular lines to find angle measures. Identify the angle relationships necessary for showing that two lines are parallel. Explore translations.

Sections 3.1 Relationships Between Lines 3.2 Theorems About Perpendicular Lines 3.3 Angles Formed by Transversals 3.4 Parallel Lines and Transversals 3.5 Showing Lines are Parallel 3.6 Using Perpendicular and Parallel Lines 3.7 Translations

Relationships Between Lines Section 3.1

Objective Identify relationships between lines.

Key Vocabulary Parallel lines Perpendicular lines Skew lines Parallel planes Lines perpendicular to a plane

Relationships Between Lines Parallel Lines – lines that are coplanar and do not intersect. Symbol - We use the symbol || to represent parallel lines, (i.e. AB || CD). A B C D

Parallel Lines Notation – use arrows on lines to indicate they are parallel. Examples:

Perpendicular Lines Lines that intersect to form right angles are perpendicularperpendicular We use the symbol ⊥ to illustrate two lines are perpendicular. ⊥ is read “ is perpendicular to.” Example:

Properties of Perpendicular Lines The following is true for all ⊥ lines: 1.⊥ lines intersect to form 4 right angles. 2. ⊥ lines intersect to form congruent adjacent angles. 3.Segments and rays can be ⊥ to lines or to other segments and rays. 4. The right angle symbol (┐) indicates that lines are ⊥.

Relationships Between Lines Indicating parallel and perpendicular lines on a diagram. Arrows are used to indicate that lines are parallel. Squares are used to indicate that lines are perpendicular (form a right angle).

Relationships Between Lines Skew Lines – lines that do not intersect and are not coplanar. (Example: lines a and b are skewed) Symbol - none a b

Relationships Between Planes Parallel Planes – two planes that do not intersect (plane ‖ plane ). A line and a plane are parallel if they do not intersect (line a ‖ plane ). a

Relationships Between Planes A Line Perpendicular to a Plane = is a line that intersects a plane in a point and that is perpendicular to every line in the plane that it intersects. Line t is perpendicular to plane A

Segments or rays are parallel, perpendicular, or skew if the lines that contain them are parallel, perpendicular, or skew. Helpful Hint

Determine whether the lines are parallel, perpendicular, or neither. a. n and m b. p and q c. n and p Lines p and q are neither parallel nor perpendicular. b. Lines n and p are perpendicular. c. SOLUTION Lines n and m are parallel. a. Example 1

Lines f and g are skew lines. b. SOLUTION Lines f and h are not skew lines because they intersect. a. Determine whether the lines are skew. a. f and h b. f and g Example 2

Name a pair of parallel lines. 1. Use the diagram. ANSWER x and y Name a pair of perpendicular lines. 2. ANSWER x and z Name a pair of skew lines. 3. ANSWER y and z

SOLUTION Plane C appears parallel to plane B. a. Name a line that appears perpendicular to plane B. b. Line l appears perpendicular to plane B. Name a plane that appears parallel to plane B. a. Example 3

Name a plane that appears parallel to plane VWX. 5. ANSWER plane RST Name a line that is perpendicular to plane VWX. 6. Think of each segment in the diagram as part of a line. ANSWER QU, TX, RQ, or ST ANSWER RV, QU, SW, or TX 4. Name a line that is skew to VW. Your Turn:

Name all planes that are parallel to plane AEF. Answer: plane BHG Example 4a:

Name all segments that intersect Answer: Example 4b:

Name all segments that are parallel to Answer: Example 4c:

Name all segments that are skew to Answer: Example 4d:

Use the figure to name each of the following. Answer: plane XYZ a. Name all planes that are parallel to plane RST. b. c. d. Answer: Your Turn:

Joke Time What did the beach say when the tide came in? Long time no sea! Why was the belt arrested? For holding up the pants. What did the dog say when it saw the pieces of a fallen tree? Bark!

Assignment Section 3.1, pg : #1-33 odd, odd, all