Download
1 / 136
1.55k likes | 2.07k Views
PARALLEL and PERPENDICULAR LINES. APPENDIX. Geometry CH 3. Relationships Between Lines. In this section we are going to learn about lines that are…. PARALLEL , PERPENDICULAR , INTERSECTING and SKEW. MENU. Geometry CH 3 Parallel & Perpendicular Lines. Relationships Between Lines.
E N D
PARALLEL and PERPENDICULAR LINES APPENDIX Geometry CH 3
Relationships Between Lines In this section we are going to learn about lines that are… PARALLEL, PERPENDICULAR, INTERSECTING and SKEW. MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines In this room, each of the walls, ceiling and floor represent a plane. Each broomstick represents a line. If both broomsticks are on the floor, Then they are in the same plane. MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines These are intersecting lines. (that just means they touch each other) Intersecting lines taped to the ceiling Intersecting lines taped to the wall MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines Intersecting Lines: Any lines that “touch” are intersecting Even though you can’t see where these two lines touch, we know that eventually they will, so they are still intersecting MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines Intersecting Lines: A special case of intersecting lines are Perpendicular lines: Perpendicular lines are lines that make right angles. And if 1 angle is a right angle, then All the angles are right angles MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines Lines that are in the same plane, and don’t touch are PARALLEL LINES MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines These lines are NOT PARALLEL. Remember, lines go on forever, If they would touch eventually, then they are NOT PARALLEL. MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines Parallel lines are always the same distance apart. MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines PARALLEL LINES are lines in the same plane that do NOT INTERSECT We usually draw them like this And include these “arrow tick marks” to indicate they are parallel. MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines m n p Which 2 lines appear to be parallel? Lines m and p. s • m//p. MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines Are the following lines parallel, intersecting or skew? c • a and c • 2. a and b a INTERSECTING b PARALLEL? Remember, they LOOK parallel, but we don’t know for sure unless we measure or are told. MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines Are the following lines parallel, intersecting or skew? f 3. k and j k PARALLEL j MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines This one has all the angles numbered Representations of PARALLEL LINES. 3-D MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines This is the room you are in right now (teacher) Can you see 2 lines that don’t touch, but aren’t parallel? MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines These lines will not touch, because they are in different planes Parallel lines must be in the same plane. MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines These are SKEW lines MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines SKEW LINES are lines that do not touch… NOT INTERSECT… NOT INTERSECT… …not because they are parallel… SKEW LINES …but because they are not in the same plane. MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines Sometimes lines are parallel, but you cannot see the plane they are on Are Lines M and L parallel? They look like they might be, but they are not in the same plane. The rule about being in the same plane includes … creating a plane L and M are parallel because they don’t intersect, and are in the same plane, even though we couldn’t see the plane. MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines Just like lines, there are PARALLEL PLANES PARALLEL PLANES Are flat surfaces that go on forever and never touch. Like the ceiling and floor in a room that goes on forever, MENU Geometry CH 3 Parallel & Perpendicular Lines
Relationships Between Lines MENU Geometry CH 3 Parallel & Perpendicular Lines
Theorems About Perpendicular Lines Perpendicular line theorems: All right angles are congruent. Anytime you see 2 right angles, they are the same shape If 2 lines are perpendicular, then they form 4 right angles. When you see 1 right angle, you know there are 4 right angles MENU Geometry CH 3 Parallel & Perpendicular Lines
Theorems About Perpendicular Lines For each of the following problems, find X. 1. 2. 3. 4. MENU Geometry CH 3 Parallel & Perpendicular Lines
Theorems About Perpendicular Lines For each of the following problems, find X. 1. 2. 3. 4. MENU Geometry CH 3 Parallel & Perpendicular Lines
Theorems About Perpendicular Lines For each of the following problems, find X. 1. 2. 3. 4. MENU Geometry CH 3 Parallel & Perpendicular Lines
Theorems About Perpendicular Lines For each of the following problems, find X. 1. 2. 3. 4. MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals When one line crosses two or more lines we call it a TRANSVERSAL MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals These two lines crossed by a TRANSVERSAL create 8 angles. 1 2 3 4 5 6 7 8 The order of the angles is not important. What is important is the relationship between the angles MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals The angles BETWEEN the two lines are called INTERIOR 1 2 3 4 5 6 7 8 Which angles are INTERIOR ANGLES? 3, 4, 5, 6 MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals The angles OUTSIDE the two lines are called EXTERIOR 1 2 3 4 5 6 7 8 Which angles are EXTERIOR ANGLES? 1,2,7,8 MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals These are EXTERIOR angles These are INTERIOR angles These are EXTERIOR angles MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals This line is called the… TRANSVERSAL MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals Which line is the transversal? L #1 #2 #3 #4 M A K B C X W S E H R MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals When we are talking about angles on the same side of the transversal, we call them… 1 2 3 4 5 6 7 8 SAME-SIDE MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals When we are talking about angles on alternate sides of the transversal, we call them… 1 2 3 4 5 6 7 8 ALTERNATE MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals Are they on the SAME SIDEor ALTERNATE sides of the transversal? MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals 3&5 4&6 Alternate Interior angles: Alternate Exterior angles: 1&7 2&8 2 1 3 4 6 5 7 8 Same side Interior angles: 3&6 4&5 2&7 1&8 Same side Exterior angles: MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals The northwest corner of this intersection CORRESPONDING ANGLES are at the same spot at the other intersection 2 1 The northwest corner of this intersection 3 These are CORRESPONDING ANGLES 7 8 5 6 4 MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals CORRESPONDING ANGLES are at the same spot at the other intersection 2 1 3 7 8 5 6 4 1 and 5 2 and 8 3 and 6 Corresponding angles: 7 and 4 MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals What is the relationship between… What is the relationship between… What is the relationship between… What is the relationship between… What is the relationship between… MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals State the relationship between the angles given. a b d c e f h g 1. a&h 2. d&f 3. c&f 4. b&h 5. c&g Same side exterior 6. b and e ?? Alternate Interior Same side Interior Alternate Exterior Corresponding MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals What if we get a more complicated picture like this? a b f e d c h g w x j k z y p m 7. f & b CORRESPONDING MENU Geometry CH 3 Parallel & Perpendicular Lines
Angles Formed by Transversals a b f e d c h g w x j k z y p m If more than 3 lines end up highlighted, the angles have NO RELATIONSHIP, 8. c & j MENU Geometry CH 3 Parallel & Perpendicular Lines
Parallel Lines and Transversals In this next section, we are going to continue focusing on this picture But, we are going to make a small change: We are going to see what happens when these lines are parallel. Activities: Measurements Using Lined Paper, Eratosthenes’ Measurement MENU Geometry CH 3 Parallel & Perpendicular Lines
Parallel Lines and Transversals SIDE NOTE: Parallel Lines and Transversals A slash through the parallel symbol || indicates the lines are not parallel. AB || CD MENU Geometry CH 3 Parallel & Perpendicular Lines
Parallel Lines and Transversals Watch what happens to corresponding angles when the lines are parallel. Click on the picture to see. MENU Geometry CH 3 Parallel & Perpendicular Lines
Parallel Lines and Transversals 1 2 POSTULATE POSTULATE Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2 MENU Geometry CH 3 Parallel & Perpendicular Lines
