3.1 Identify Pairs of Lines and Angles 3.2 Use Parallel Lines and Transversals Objectives: 1.To differentiate between parallel, perpendicular, and skew lines 2.To compare Euclidean and Non-Euclidean geometries 3.To identify a transversal and various angle pairs 4.To find angle pair measurements with parallel lines cut by a transversal

Vocabulary Parallel LinesSkew Lines Perpendicular Lines Transversal As a group, define each of these without your book. Draw a picture for each word and leave a bit of space for additions and revisions.

Example 1 Use the diagram to answer the following. 1.Name a pair of lines that intersect. 2.Would JM and NR ever intersect? 3.Would JM and LQ ever intersect?

Parallel Lines parallel lines Two lines are parallel lines if and only if they are coplanar and never intersect. The red arrows indicate that the lines are parallel.

parallel lines Two lines are parallel lines if and only if they are coplanar and never intersect. Parallel Lines

skew lines Two lines are skew lines if and only if they are not coplanar and never intersect. Skew Lines

Example 2 Think of each segment in the figure as part of a line. Which line or plane in the figure appear to fit the description? 1.Line(s) parallel to CD and containing point A. 2.Line(s) skew to CD and containing point A.

Example 2 3.Line(s) perpendicular to CD and containing point A. 4.Plane(s) parallel plane EFG and containing point A.

Example 3 Draw line ℓ and point P. How many lines can you draw through point P that are parallel to line ℓ? How many lines can you draw through point P that are perpendicular to line ℓ?

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.

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. Euclid’s Fifth Postulate Also referred to as Euclid’s Fifth Postulate

Euclid’s Fifth Postulate Some mathematicians believed that the fifth postulate was not a postulate at all, that it was provable. So they assumed it was false and tried to find something that contradicted a basic geometric truth.

Example 4 If the Parallel Postulate is false, then what must be true? Hyperbolic Geometry Hyperbolic Geometry 1.Through a given point not on a given line, you can draw more than one line parallel to the given line. This makes Hyperbolic Geometry.Hyperbolic Geometry

Example 4 If the Parallel Postulate is false, then what must be true? Hyperbolic Geometry Hyperbolic Geometry 1.Through a given point not on a given line, you can draw more than one line parallel to the given line. This makes Hyperbolic Geometry.Hyperbolic Geometry

Example 4 If the Parallel Postulate is false, then what must be true? Elliptic Geometry Elliptic Geometry 2.Through a given point not on a given line, you can draw no line parallel to the given line. This makes Elliptic Geometry.Elliptic Geometry

Great Circles Great Circle: Great Circle: The intersection of the sphere and a plane that cuts through its center. Think of the equator or the Prime Meridian The lines in Euclidean geometry are considered great circles in elliptic geometry. Great circles divide the sphere into two equal halves. l

Example 5 1.In Elliptic geometry, how many great circles can be drawn through any two points? 2.Suppose points A, B, and C are collinear in Elliptic geometry; that is, they lie on the same great circle. If the points appear in that order, which point is between the other two?

Transversal transversal A line is a transversal if and only if it intersects two or more coplanar lines. –When a transversal cuts two coplanar lines, it creates 8 angles, pairs of which have special names

Transversal corresponding angles  1 and  5 are corresponding angles alternate interior angles  3 and  6 are alternate interior angles alternate exterior angles  1 and  8 are alternate exterior angles consecutive interior angles  3 and  5 are consecutive interior angles

Example 5 Classify the pair of numbered angles.

Investigation 1 Use the following Investigation to help you complete some postulates and theorems about parallel lines and a transversal. You will need notebook and patty papers.

Investigation 1 1.Using the lines on a piece of paper as a guide, draw a pair of parallel lines. Now draw a transversal that intersects the parallel lines. Label the angles with numbers.

Investigation 1 2.Place a patty paper over the set of angles  1,  2,  3, and  4 and copy the two intersecting lines onto the patty paper. 3.Slide the patty paper down and compare angles 1 through 4 with angles 5 through 8.

Investigation 1 Do you notice a relationship between pairs of corresponding, alternate interior, and alternate exterior angles? Let’s make a four window foldable to summarize each of the following postulates and theorems

Four Window Foldable Start by folding a blank piece of paper in half lengthwise, and then folding it in half in the opposite direction. Now fold it in half one more time in the same direction.

Four Window Foldable Now unfold the paper, and then while holding the paper vertically, fold down the top one-fourth to meet the middle. Do the same with the bottom one- fourth.

Four Window Foldable To finish your foldable, cut the two vertical fold lines to create four windows. Outside: Outside: Name Inside Flap: Inside Flap: Illustration Inside: Inside: Postulate or Theorem

Four Window Foldable Corresponding Angles Postulate If two parallel lines are cut by a transversal, then pairs of corresponding angles are congruent. Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then pairs of alternate interior angles are congruent.

Four Window Foldable Alternate Exterior Angle Theorem If two parallel lines are cut by a transversal, then pairs of alternate exterior angles are congruent. Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then pairs of consecutive interior angles are supplementary.

Example 6: SAT In the figure, if l 1 || l 2 and l 3 || l 4, what is y in terms of x.

Example 7: SAT In the figure, if l || l, what is the value of x ?

Example 8 Prove the Alternate Interior Angle Theorem.

Example 9 Calculate each lettered angle measure.

Example 10 Find the values of x, y, and z if k || l || m.