1. 11 Chapter Introductory Geometry
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2. NCTM Standard: Geometry
In grades preK–2, all students should
recognize, name, build, draw, compare, and sort two- and three-dimensional shapes;
describe attributes and parts of two-and three-dimensional shapes;
investigate and predict the results of putting together and taking apart two- and three-dimensional shapes. (p. 96)
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3. NCTM Standard: Geometry
In grades 3–5 all students should
sort, build, draw, model, trace, measure, and construct, the capacity to visualize geometric relationships
learn to reason and to make, test, and justify conjectures
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4. NCTM Standard: Geometry
In grades 6–8, all students should
precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties;
use visual tools such as networks to represent and solve problems. (p. 240)
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11-1 Basic Notions
Linear Notions
Planar Notions
Other Planar Notions
Angles
Angle Measurement
Types of Angles
Perpendicular Lines
A Line Perpendicular to a Plane
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6. Undefined terms: points, lines, and planes
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Lines
A line has no thickness and it extends forever in two directions.
Given two points, there is one and only one line that connects these points.
If P and Q are any two points, we can create a number line on the line PQ such that there is one-to-one correspondence between the points on the line and real numbers.
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Linear Notions
Collinear points
Line ℓ contains points A, B, and C.
Points A, B, and C belong to line ℓ.
Points A, B, and C are collinear.
Points A, B, and D are not collinear.
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Linear Notions
Between
Point B is between points A and C on line ℓ.
Point D is not between points A and C.
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Linear Notions
AB or BA
Line segment
A subset of a line that contains two points of the line and all points between those two points.
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Linear Notions AB
Ray
A subset of the line AB that contains the endpoint A, the point B, all points between A and B, and all points C on the line such that B is between A and C.
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Planar Notions
Coplanar
Points D, E, and G are coplanar.
Points D, E, F, and G are not coplanar.
Lines DE, DF, and FE are coplanar.
Lines DE and EG are coplanar.
Lines DE and EG are intersecting lines; they intersect at point E.
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Planar Notions
Skew lines
Lines GF and DE are skew lines. They do not intersect, and there is no plane that contains them.
Concurrent lines
Lines DE, EG, and EF are concurrent lines; they intersect at point E.
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Planar Notions
Parallel lines
Line m is parallel to line n. They have no points in common.
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15. Axioms About Points, Lines, and Planes
There is exactly one line that contains any two distinct points.
If two points lie in a plane, then the line containing the points lies in the plane.
If two distinct planes intersect, then their intersection is a line.
There is exactly one plane that contains any three distinct noncollinear points.
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16. Theorems About Points, Lines, and Planes
A line and a point not on the line determine a plane.
Two parallel lines determine a plane.
Two intersecting lines determine a plane.
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Other Planar Notions
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Half plane
Line AB separates plane  into two half-planes.
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Angles
Angle – formed by two rays with the same endpoint.
Sides of an angle – the two rays that form an angle.
Vertex – the common endpoint of the two rays that form an angle.
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Angles
Adjacent angles – two angles with a common vertex and a common side, but without overlapping interiors.
QPR is adjacent to RPS.
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Angle Measurement
Degree : of a rotation about a point
Minute : of a degree
Second : of a minute
Protractor
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Example 11-1
a. Find the measure of BAC if m1 = 47°45′ and m2 = 29°58′ .
mBAC = 17°47′
b. Express 47°45′36″ as a number of degrees.
47°45′36″ = 47°+ 0.75° ° = 47.76°
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Types of Angles
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Types of Angles
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Perpendicular Lines
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26. A Line Perpendicular to a Plane
A line perpendicular to a plane is a line that is perpendicular to every line in the plane through its intersection with the plane.
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Perpendicular Planes
If P is any point on AD, Q in plane α, and S in plane β so that
PQ  AD and PS  AD, then QPS is also a right angle.
Since QPS measures 90°, the planes are perpendicular.
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Dihedral Angles
A dihedral angle is formed by the union of two half-planes and the common line defining the half-planes.
A dihedral angle is measured by any of the associated planar angles such OPD, where PO  AC and PD  AC.
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