Parallel and Perpendicular Lines

Today’s Learning Goals We will investigate the properties of angle measurements when you have perpendicular or parallel lines.

Definitions Consider the following two lines: These two lines are called parallel lines. Parallel lines are lines that never meet. How many lines are parallel to the two lines above? Nice…an infinite amount of parallel lines. The arrows at either end signify that lines go on forever in both directions.

Notation for Lines When given more than one line (like below), we need some way of distinguishing them apart. Similar to what we did with angles, sometimes we put points on the lines and use the letters to describe lines. A B C D How could we name the top line? How could we name the bottom line? Yes…AB Good…CD Notice how there are arrows in both directions to denote a line.

Line Segments vs. Lines Line segments are only parts of a line. A B C D What are the two line segments shown above? Yes…AB and CD. What is the difference between how we denoted a line from a line segment? Great…Lines have a double arrow over the letters while a line segment does not have the arrows.

Rays vs. Line Segments and Lines Rays are lines that only go in one direction from a starting point. A B C D Using the starting point to name the rays, what are some rays shown above? What is the difference between how we denoted a ray from a line segment? Great…Line segments needed two letters while rays only need the starting point and rays have single arrows. Excellent…A, B, C, D, A, B, C, and D.

Notation for Lines Another way to denote a line is to give it one letter to the side of the line. Above, we could say we have parallel lines p and q. The symbol “||” means parallel. So, p || q. p q A transversal is a line that intersects two or more parallel lines. Line t is a transversal above. t

Parallel Lines When a transversal intersects two or more parallel lines, we create angles & 3 2 & 4 5 & 7 6 & 8 Corresponding angles are angles that are on the same side of the transversal and same side of the parallel lines. Which angles are corresponding in the figure below? p q t

Parallel Lines Imagine dropping one of the parallel lines down and plopping them one on top of the other. When we plop line p onto line q,  1 plops on top of which angle? p q t Yes…  1 plops onto  3. If  1 plops perfectly on top of  3, what does that mean about the angle measurements? Nice…m  1 = m  3.

Parallel Lines We just saw that when we plop line p on top of line q, m  1 = m  3. What other angles will plop on top of each other? p q t Great…  1 plops onto  3  2 plops onto  4  8 plops onto  6  7 plops onto  5

Parallel Lines If corresponding angles plop on top of each other perfectly, what does that mean about corresponding angles when we have parallel lines? p q t Excellent…corresponding angles have the same measurement (are congruent).

Definitions Now, consider these: These lines are called perpendicular lines. Perpendicular lines are lines that form a 90  angle between them. A square between the lines is used to denote the 90 .

Perpendicular Lines The symbol  means “perpendicular”. If the lines above are labeled as shown, how would we say which lines are perpendicular? s t L M P Q Nice…s  t and

Construction Parallel and perpendicular line segments are often used in construction, art, and other real-world applications. Why do you think I said parallel and perpendicular line segments are often used instead of parallel and perpendicular lines? Nice…lines go on forever in both directions and that is impossible in most real-world applications.

Equal Angles with Parallel Lines When we have parallel lines with a transversal cutting through them, there are some interesting properties of equality for the angles that are made. Consider the following. Which angles did we say are equal if p || q? p q t Nice…all corresponding angles are equal. So,  1 =  3,  2 =  4,  8 =  6,  7 =  5.

Equal Angles with Parallel Lines What angles are also equal because they are vertical angles? Nice…  1 =  7,  2 =  8,  3 =  5, and  4 =  p q t

Equal Angles with Parallel Lines p q t We said that 1 = 3 and 1 = 7. What does that mean about angles 3 and 7? Great…  3 =  7.  3 and  7 are called alternate interior angles because they are on different sides of the transversal and both between (inside) the parallel lines.

Equal Angles with Parallel Lines There is another pair of alternate interior angles different than angles 3 and 7. What are they? Yes…  2 and  6 are alternate interior angles p q t Is  2 =  6? Explain how you know. Excellent…  2 =  6 because  2 =  4 and  4 =  6.

Partner Work You have 15 minutes to work on the following problems with your partner.

For those that finish early 1. Look around the room. Explain where you see each of these in the classroom. a) parallel line segments b) perpendicular line segments c) transversals d) corresponding angles 2. Is there such a thing as parallel lines or perpendicular lines in the real world? Explain.

For those that finish early Consider the following diagram where p || q || r. Determine the measures of all 19 missing angles. p q r 50  60  110  11 22 33 44 55 66 77 88 99  10  11  12  13  14  15  16  17  18  19

Big Idea from Today’s Lesson Corresponding Angles are congruent. Alternate interior angles are congruent. Alternate exterior angles are congruent.

Homework Pgs. 220 – 221 (1 – 20)