1-2 (For help, go to the Skills Handbook, page 722.) 1. y = x + 52. y = 2x – 4 3. y = 2x y = –x...
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Transcript of 1-2 (For help, go to the Skills Handbook, page 722.) 1. y = x + 52. y = 2x – 4 3. y = 2x y = –x...
1-2
(For help, go to the Skills Handbook, page 722.)
1. y = x + 5 2. y = 2x – 4 3. y = 2x
y = –x + 7 y = 4x – 10 y = –x + 15
4. 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 1-2GEOMETRY LESSON 1-2
Points, Lines, and PlanesPoints, 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 1-2GEOMETRY LESSON 1-2
Points, Lines, and PlanesPoints, 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 1-1GEOMETRY LESSON 1-1
Patterns and Inductive ReasoningPatterns 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 1-1GEOMETRY LESSON 1-1
Patterns and Inductive ReasoningPatterns 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 1-1GEOMETRY LESSON 1-1
Patterns and Inductive ReasoningPatterns 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 1-1GEOMETRY LESSON 1-1
Patterns and Inductive ReasoningPatterns and Inductive Reasoning
1-1
Definitions
-All points that lie on the same line are said to be collinear
Example:
Collinear points
GEOMETRY LESSON 1-1GEOMETRY LESSON 1-1
Patterns and Inductive ReasoningPatterns 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 1-2GEOMETRY LESSON 1-2
Points, Lines, and PlanesPoints, 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 1-1GEOMETRY LESSON 1-1
Patterns and Inductive ReasoningPatterns 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 1-1GEOMETRY LESSON 1-1
Patterns and Inductive ReasoningPatterns 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 1-1GEOMETRY LESSON 1-1
Patterns and Inductive ReasoningPatterns 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 1-2GEOMETRY LESSON 1-2
Points, Lines, and PlanesPoints, Lines, and Planes
1-2
Points and lines on the same plane are said to be coplanar
Coplanar
GEOMETRY LESSON 1-1GEOMETRY LESSON 1-1
Patterns and Inductive ReasoningPatterns 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 1-2GEOMETRY LESSON 1-2
Points, Lines, and PlanesPoints, 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 1-2GEOMETRY LESSON 1-2
Points, Lines, and PlanesPoints, Lines, and Planes
1-2
Through any two points, there is exactly ONE line
Postulate 1-1:
GEOMETRY LESSON 1-1GEOMETRY LESSON 1-1
Patterns and Inductive ReasoningPatterns 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 1-2GEOMETRY LESSON 1-2
Points, Lines, and PlanesPoints, 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 1-2GEOMETRY LESSON 1-2
Points, Lines, and PlanesPoints, Lines, and Planes
1-2