3-1 and 3-2: Parallel Lines and Transversals Mr. Schaab’s Geometry Class Our Lady of Providence Jr.-Sr. High School 2014-2015
Identifying Pairs of Lines Two lines are: Parallel if they do not intersect and are coplanar. Perpendicular if they intersect to form right angles. Skew if they do not intersect and are not coplanar. (Note: ALL intersecting lines are coplanar!)
Identifying Pairs of Lines - Example In the cube below, identify the following: A pair of perpendicular lines: Line PS and Line PW A pair of parallel lines: Line PW and Line QX Line PW and Line RY Line QX and Line RY A pair of skew lines: Line PS and Line QX Line PS and Line RY
Identifying Pairs of Planes - Example In the cube below, identify the following: A pair of perpendicular planes: Plane SRY and Plane PWZ Plane PQX and Plane QXY A pair of Parallel planes: Plane PQX and Plane SRY Plane PWZ and Plane QRY Plane PQR and WXY Pair of skew planes: None!
Angles and Transversals Transversal – a line that intersects two or more coplanar lines at different points. In the diagram on the right, line t is a the transversal of lines L1 and L2. A transversal that intersects two lines forms 8 angles, all of which have special relationships.
Angle Relationships Corresponding Angles Two angles that are in matching locations on different intersections. ∠1 and ∠5 are corresponding angles. 1 5
Angle Relationships Alternate Interior Angles Two angles that lie between the two lines and on opposite sides of the transversal. ∠4 and ∠5 are alternate interior angles. 4 5
Angle Relationships Alternate Exterior Angles Two angles that lie outside the two lines and on opposite sides of the transversal. ∠2 and ∠7 are alternate exterior angles. 2 7
Angle Relationships Consecutive Interior Angles Two angles that lie between the two lines and on the same side of the transversal. These are also called “Same-side interior angles.” ∠3 and ∠5 are consecutive interior angles. 3 5
Angles and Transversals - Example Identify all pairs of angles of the given type: Corresponding: ∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8 Alternate Interior: ∠2 & ∠7, ∠4 & ∠5 Alternate Exterior: ∠3 & ∠6, ∠1 & ∠8 Consecutive Interior: ∠4 & ∠7, ∠2 & ∠5 6 5 8 7 2 1 4 3
Parallel Lines and Transversals Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then all pairs of corresponding angles are congruent. ∠1 ≅ ∠5 1 5
Parallel Lines and Transversals Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then all pairs of alternate interior angles are congruent. ∠4 ≅ ∠5 4 5
Parallel Lines and Transversals Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then all pairs of alternate exterior angles are congruent. ∠1 ≅ ∠8 1 8
Parallel Lines and Transversals Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then all pairs of consecutive interior angles are supplementary. m∠3 + m∠5 = 180° 3 5
Parallel Lines and Transversals If you’re angle 3, then you have a lot of relationships! The other angles must really like you. ∠3 & ∠1 – Linear Pair (supplementary) ∠3 & ∠2 – Vertical Angles (congruent) ∠3 & ∠4 – Linear Pair (supplementary) ∠3 & ∠5 – Consecutive Interior Angles (supplementary) ∠3 & ∠6 – Alternate Interior Angles (congruent) ∠3 & ∠7 – Corresponding Angles (congruent) ∠3 & ∠8 – No relationship (∠8 is a jerk.) 1 2 3 4 5 6 7 8