1-2 (For help, go to the Skills Handbook, page 722.) 1. y = x y = 2x – 4 3. y = 2x y = –x + 7 y = 4x – 10 y = –x Copy the diagram of the four points A, B, C, and D. Draw as many different lines as you can to connect pairs of points. GEOMETRY LESSON 1-2 Points, Lines, and Planes Solve each system of equations.

1. By substitution, x + 5 = –x + 7; adding x – 5 to both sides results in 2x = 2; dividing both sides by 2 results in x = 1. Since x = 1, y = (1) + 5 = 6. (x, y) = (1, 6) 2. By substitution, 2x – 4 = 4x – 10; adding –4x + 4 to both sides results in –2x = –6; dividing both sides by –2 results in x = 3. Since x = 3, y = 2(3) – 4 = 6 – 4 = 2. (x, y) = (3, 2) 3. By substitution, 2x = –x + 15; adding x to both sides results in 3x = 15; dividing both sides by 3 results in x = 5. Since x = 5, y = 2(5) = 10. (x, y) = (5, 10) 4. The 6 different lines are AB, AC, AD, BC, BD, and CD. Solutions GEOMETRY LESSON 1-2 Points, Lines, and Planes 1-2

-A single location in space -Indicated by an “infinitely small” dot -Named by a capital letter Example: Point GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1 Definitions Good to Know: -A geometric figure is a set of points -Space is defined as the set of all points

-A series of points that extend infinitely in two opposite directions -Drawn with an arrow at each end, indicating that it extends indefintely Example: Line GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1 Definitions

A line can be named either one of two ways: 1. A line can be named by placing the line symbol ( ) over any two points that fall on the line. Example: Line GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1 Definitions

A line can be named either one of two ways: 2. A line can be named with a single, lowercase letter Example: Line GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1 Definitions

-All points that lie on the same line are said to be collinear Example: Collinear points GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1 Definitions -Points C, A and B are collinear -Points C, A and D are NOT collinear -Are points D, B and E collinear? -YES!

Any other set of three points do not lie on a line, so no other set of three points is collinear. For example, X, Y, and Z and X, W, and Z form triangles and are not collinear. In the figure below, name three points that are collinear and three points that are not collinear. Points Y, Z, and W lie on a line, so they are collinear. GEOMETRY LESSON 1-2 Points, Lines, and Planes 1-2

-A flat surface with no thickness -Comprised of an infinite number of lines -Extends without end in the direction of all of its lines Example: Planes GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1 Definitions

A plane can be named in either of two ways: 1. A single capital letter Example: Planes GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1 Definitions

A plane can be named in either of two ways: 2. the combination of three of its non-collinear points Example: Planes GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1 Definitions

You can name a plane using any three or more points on that plane that are not collinear. Some possible names for the plane shown are the following: plane RST plane RSU plane RTU plane STU plane RSTU Name the plane shown in two different ways. GEOMETRY LESSON 1-2 Points, Lines, and Planes 1-2

Points and lines on the same plane are said to be coplanar Coplanar GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1 Definitions -line b, line c, and point K are coplanar -line a and line b are NOT coplanar -line a and point K are coplanar, but not on the plane shown!

As you look at the cube, the front face is on plane AEFB, the back face is on plane HGC, and the left face is on plane AED. The back and left faces of the cube intersect at HD. Planes HGC and AED intersect vertically at HD. Use the diagram below. What is the intersection of plane HGC and plane AED? GEOMETRY LESSON 1-2 Points, Lines, and Planes 1-2

Points X, Y, and Z are the vertices of one of the four triangular faces of the pyramid. To shade the plane, shade the interior of the triangle formed by X, Y, and Z. Shade the plane that contains X, Y, and Z. GEOMETRY LESSON 1-2 Points, Lines, and Planes 1-2

Through any two points, there is exactly ONE line Postulate 1-1: GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1 Postulates Postulate 1-2: If two lines intersect, then they intersect in exactly ONE point. Postulate 1-3: If two planes intersect, then they intersect in exactly ONE line. Postulate 1-4 Through any three noncollinear points, there is exactly one plane.

1. Name three collinear points. 2. Name two different planes that contain points C and G. 3. Name the intersection of plane AED and plane HEG. 4. How many planes contain the points A, F, and H? 5. Show that this conjecture is false by finding one counterexample: Two planes always intersect in exactly one line. Use the diagram at right. GEOMETRY LESSON 1-2 Points, Lines, and Planes 1-2

1. Name three collinear points. 2. Name two different planes that contain points C and G. 3. Name the intersection of plane AED and plane HEG. 4. How many planes contain the points A, F, and H? 5. Show that this conjecture is false by finding one counterexample: Two planes always intersect in exactly one line. Use the diagram at right. D, J, and H planes BCGF and CGHD HE 1 Sample: Planes AEHD and BFGC never intersect. GEOMETRY LESSON 1-2 Points, Lines, and Planes 1-2