3.1 PARALLEL LINES Thompson How can I recognize planes, transversals, pairs of angles formed by a transversal, and parallel lines?
What You Should Learn Why You Should Learn It Goal 1: How to identify angles formed by two lines and a transversal Goal 2: How to use properties of parallel lines
Angles Formed by a Transversal Transversal – a line that intersects two or more coplanar lines at different points In the figure, the transversal t intersects the lines L and M t L M
Corresponding Angles Two angles are corresponding angles if they occupy corresponding positions, such as t L M
Alternate Interior Angles Two angles are alternate interior angles if they lie between L and M on opposite sides of t, such as t L M
Alternate Exterior Angles Two angles are alternate exterior angles if they lie outside L and M on opposite sides of t, such as t L M
Same-Side Interior Angles (consecutive) Two angles are same-side interior angles if they lie between L and M on the same side of t, such as t L M
These definitions depend on what line is the transversal. Before you define the angle terms, you need to think about which line is the transversal! Even within the same diagram, the transversal can change. This will often lead to “ignoring” other parts of the diagram (as we’ll see next…)
Example 1 Naming Pairs of Angles How is related to the other angles? n m L
Example 1 Naming Pairs of Angles How is related to the other angles? are a linear pair. So are are vertical angles are alternate exterior angles. So are are corresponding angles. So are n m L
PARALLEL LINES AND CONGRUENT ANGLES While these definitions apply whenever two lines are cut by a transversal, we will normally talk about this for parallel lines 1. What is the Parallel Postulate and how can I use it? 2. What are the pairs of congruent angles formed by parallel lines cut by a transversal? 3. Lets reexamine the angle definitions for parallel lines
P ARALLEL L INES Two coplaner lines that do not intersect The book notes them with “matching arrows” see page 117
TRANSVERSAL (Parallel) A line that intersects two coplanar lines in two distinct points.
SPECIAL ANGLES (parallel) Interior Angles – lie between the two lines ( 3, 4, 5, and 6) Alternate Interior Angles – are on opposite sides of the transversal. ( 3 & 6 AND 4 and 5) Same-Side Interior Angles – are on the same side of the transversal. ( 3 & 5 AND 4 & 6)
M ORE S PECIAL A NGLES Exterior Angles – lie outside the two lines ( 1, 2, 7, and 8) Alternate Exterior Angles – are on opposite sides of the transversal ( 1& 8 AND 2 & 7) Corresponding Angles – same location, different intersections ( 2 & 6, 4 & 8, 1 & 5, 3 & 7)
P ARALLEL L INES AND A T RANSVERSAL 1. On your paper, construct two parallel lines, then construct an “angled” transversal. 2.Label each angle made ( 1 - 8) 3.Based on appearance, make a conjecture as to which angles are congruent, supplementary, or complementary to each other. 4.Using the protractor, measure and label each angle. 5.Make 4 conjectures about the angle pairs: ( alternate int, consecutive ext, corresponding…)
I F 2 PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN ANY PAIR OF THE ANGLES FORMED ARE EITHER CONGRUENT OR SUPPLEMENTARY.
Vertical Angles & Linear Pair Vertical Angles: Linear Pair: 1 4, 2 3, 5 8, 6 7 Two angles that are opposite angles. Vertical angles are congruent. 1 & 2, 2 & 4, 4 & 3, 3 & 1, 5 & 6, 6 & 8, 8 & 7, 7 & 5 Supplementary angles that form a line (sum = 180 )
Angles and Parallel Lines If two parallel lines are cut by a transversal, then the following pairs of angles are congruent. 1. Corresponding angles 2. Alternate interior angles 3. Alternate exterior angles If two parallel lines are cut by a transversal, then the following pairs of angles are supplementary. 1. Same-Side interior angles 2. Same-Side exterior angles Continued…..
Corresponding Angles & Consecutive Angles Corresponding Angles: Two angles that occupy corresponding positions. 2 6, 1 5, 3 7, 4
Consecutive Angles Consecutive Interior Angles: Two angles that lie between parallel lines on the same sides of the transversal. Consecutive Exterior Angles: Two angles that lie outside parallel lines on the same sides of the transversal. m 3 +m 5 = 180º, m 4 +m 6 = 180º m 1 +m 7 = 180º, m 2 +m 8 = 180º
Alternate Angles Alternate Interior Angles: Two angles that lie between parallel lines on opposite sides of the transversal (but not a linear pair). Alternate Exterior Angles: Two angles that lie outside parallel lines on opposite sides of the transversal. 3 6, 4 5 2 7, 1
Example: If line AB is parallel to line CD and s is parallel to t, find the measure of all the angles when m< 1 = 100°. Justify your answers. t s D C B A
Example: 1. the value of x, if m<3 = 4x + 6 and the m<11 = 126. If line AB is parallel to line CD and s is parallel to t, find: 2. the value of x, if m<1 = 100 and m<8 = 2x the value of y, if m<11 = 3y – 5 and m<16 = 2y t s D C B A
I F TWO LINES ARE P ARALLEL : 1. A pair of corresponding s are . 2. A pair of alternate interior s are . 3. A pair of alternate exterior s are . 4. A pair of consecutive interior s are supplementary. 5. A pair of consecutive exterior s are supplementary.
I NVESTIGATING P OSTULATES 1.Construct a line named AB. 2.Somewhere above or below the line, put a pt P. 3.Construct 2 lines that go through pt P that are also || to AB
Parallel Postulates The following have been proven true and can be applied to parallel lines.
T HROUGH A POINT NOT ON A LINE THERE IS EXACTLY ONE PARALLEL TO THE GIVEN LINE. Parallel Postulates
I NVESTIGATING P OSTULATES 1.Construct 2 parallel lines AB and CD. 2.Construct LP so that it is to AB and also passes through CD. 3.Measure the angles of the intersection on LP and CD. 4.What can you conclude?
I N A PLANE, IF A LINE IS PERPENDICULAR TO 1 OF 2 PARALLEL LINES, THEN IT IS PERPENDICULAR TO THE OTHER.
I F 2 LINES ARE PARALLEL TO A 3 RD LINE, THEN THEY ARE PARALLEL TO EACH OTHER. ( TRANSITIVE PROP. OF || LINES )
Classwork - Go! Starting on Pg