Section1 3a
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This document defines several key geometry terms: 1) A segment is defined as two points on a line connected by all points between them. 2) A ray is one endpoint point on a line extending in one direction. 3) Opposite rays extend in opposite directions from a common endpoint along the same line. 4) Length refers to the distance between two points, which is the absolute value of the difference between their coordinates on a number line.
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The document defines and provides examples of basic geometry terms including:
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This document defines and discusses various geometric concepts including:
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1. Points, lines, and planes are basic geometric objects. A point has no size or dimensions. A line extends indefinitely in both directions and has one dimension. A plane extends indefinitely and has two dimensions.
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This document defines key geometry concepts such as points, lines, planes, and their relationships. It provides examples of naming points, lines, and planes, including collinear points that lie on the same line and coplanar points that lie in the same plane. Examples also demonstrate naming segments and rays with different endpoints, and identifying opposite rays. Diagrams show intersecting lines and planes, including lines within a plane, lines that do not intersect a plane, and lines intersecting a plane at a point. Two intersecting planes are shown meeting at a line of intersection. Guided practice problems apply the concepts to name intersections and identify relationships in diagrams.
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This document defines and provides examples of basic geometric terms including points, lines, planes, segments, rays, and intersections. It begins by stating that points, lines, and planes are undefined terms that are intuitive ideas used to define other terms. It then defines a point as having no size, a line as made up of points extending in two directions, and a plane as a flat surface that extends indefinitely. Several examples are provided to demonstrate collinear points, coplanar points, line segments, rays, and intersections between geometric figures. Homework problems involving these concepts are listed at the end.
GEOMETRY: POINTS, LINES. PLANE
By this end of the presentation you will be able to:
Identify and model points, lines, and planes.
Identify collinear and coplanar points.
Identify non collinear and non coplanar points.
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1. Points, lines, and planes are basic geometric objects. A point has no size or dimensions. A line extends indefinitely in both directions and has one dimension. A plane extends indefinitely and has two dimensions.
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This document defines key geometry concepts such as points, lines, planes, and their relationships. It provides examples of naming points, lines, and planes, including collinear points that lie on the same line and coplanar points that lie in the same plane. Examples also demonstrate naming segments and rays with different endpoints, and identifying opposite rays. Diagrams show intersecting lines and planes, including lines within a plane, lines that do not intersect a plane, and lines intersecting a plane at a point. Two intersecting planes are shown meeting at a line of intersection. Guided practice problems apply the concepts to name intersections and identify relationships in diagrams.
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This document defines and provides examples of basic geometric terms including points, lines, planes, segments, rays, and intersections. It begins by stating that points, lines, and planes are undefined terms that are intuitive ideas used to define other terms. It then defines a point as having no size, a line as made up of points extending in two directions, and a plane as a flat surface that extends indefinitely. Several examples are provided to demonstrate collinear points, coplanar points, line segments, rays, and intersections between geometric figures. Homework problems involving these concepts are listed at the end.
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By this end of the presentation you will be able to:
Identify and model points, lines, and planes.
Identify collinear and coplanar points.
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- Parallelogram: Area = base x height
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The document outlines several postulates about segments and their measures on a number line, including that the distance between two points can be found by taking the absolute value of the difference between their coordinates, and that if a point B is between points A and C, then the sum of the lengths of segments AB and BC will equal the length of segment AC. Examples are provided to demonstrate applying these postulates, such as using the distance formula or Pythagorean theorem to find lengths of segments. Key terms like congruent segments are also defined.
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- A plane is a flat, thickness-less surface that extends indefinitely in all directions and is usually pictured as a four-sided shape.
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- It states theorems about tangents, including that a tangent line is perpendicular to the radius at the point of tangency, and a line perpendicular to the radius at a point on the circle is a tangent.
- It provides examples demonstrating identifying special segments and lines related to circles, identifying common tangents, and using properties of tangents to solve problems.
10.1 tangents to circles
The document summarizes key definitions and properties related to circles and tangents to circles:
- It defines circles, radii, diameters, chords, secants, and tangents. It also defines tangent circles and common tangents.
- It states theorems about tangents, including that a tangent line is perpendicular to the radius at the point of tangency, and a line perpendicular to the radius at a point on the circle is a tangent.
- It gives examples demonstrating identifying special segments and lines related to circles, identifying common tangents, and using properties of tangents to solve problems.
1.2 points _lines____planes
The document defines basic geometric terms including points, which have no dimensions; lines, which extend infinitely in one direction; and planes, which extend infinitely in two dimensions. It also defines line segments as portions of lines between two endpoints, rays as extending infinitely from one endpoint, and discusses the intersection of lines and planes. The examples provide visual representations of these geometric concepts and their relationships to one another.
AXIOMATIC STRUCTURE.pptx mathematics g88
This document discusses axiomatic systems and their properties. It begins by defining an axiomatic system as a logical system with an explicitly stated set of axioms from which theorems can be derived. It then explains key aspects of axiomatic systems including:
- Axioms serve as the starting points and basic truths, while theorems are statements deduced from the axioms.
- Important properties of axiomatic systems are consistency, independence, and completeness. Consistency means no contradictions, independence means axioms can't be derived from others, and completeness means every statement is provable or disprovable.
- Examples are given of Euclid's common notions and post
Maths sa 2 synopsis
The document provides information about topics in grade 9 math, including:
1) Linear equations in two variables and their graphical representations as lines.
2) Properties of quadrilaterals such as parallelograms, and how to classify them.
3) Finding areas of parallelograms, triangles, and how these shapes are related.
4) Properties of circles such as angles subtended by chords and arcs.
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This document provides an introduction to direction cosines and direction ratios of lines in three-dimensional space, as well as equations of lines in space. It defines direction angles and explains how to calculate direction cosines from these angles. It also defines direction ratios and shows how they relate to direction cosines. The document gives examples of finding direction cosines and checking if points are collinear using direction ratios. It then derives equations for lines passing through a point and parallel to a vector, and lines passing through two given points, in both vector and Cartesian forms. Examples are provided to illustrate converting between forms.
10.1 tangents to circles
This document provides definitions and examples related to circles and tangents. It defines key terms like radius, diameter, chord, secant, and tangent. Examples demonstrate identifying these segments and determining if lines are tangent to circles. Theorems are presented about properties of tangents, such as tangents being perpendicular to radii and two tangents from the same exterior point being congruent. Proofs of theorems are also provided. Exercises apply these concepts, like using properties of tangents to find missing values.
A non-Euclidean model
1) The document presents a non-Euclidean model called the Poincare disk model, which represents hyperbolic geometry in the unit disk.
2) It then develops this model by defining lines called psi-lines that cut the unit disk at an angle psi. This creates a psi-model of hyperbolic geometry.
3) The document lists eight theorems that are valid in the special case of the zero-model, where lines are tangent to the unit disk. It states these theorems can be translated into Euclidean geometry theorems.
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Section1 5
This document contains 7 statements about geometry concepts related to points, lines, and planes. Students are asked to classify each statement as True or False. The statements cover topics such as the number of planes a triangle can lie in, properties of points and lines, how many points define a plane, and how many planes can pass through given lines and points.
Section1 4
The document introduces several key concepts about angles:
1) It defines an angle as the union of two rays with a common endpoint called the vertex.
2) It classifies angles as acute, right, or obtuse based on their measure.
3) It defines congruent angles, adjacent angles, and a linear pair of angles.
4) It introduces the angle bisector, angle addition postulate, and linear pair postulate as tools for working with angles.
Section1 2a
The document defines the basic terms in geometry:
1) A point has 0 dimensions and represents a location in space.
2) A line extends in one direction and has 1 dimension (length).
3) A plane extends indefinitely in two directions and has 2 dimensions (length and width).
It also defines related terms like space, collinear points, coplanar points, and intersections. The document concludes with examples to classify as true or false.
Section1 3b
The document introduces the Segment Addition Postulate, which states that if point B is between points A and C, then the sum of the lengths of segments AB and BC equals the length of segment AC. It then provides an example of using this postulate to solve for the variable x in a geometry problem where one segment bisects another.
Section2 4
The document discusses three kinds of special angle pairs: complementary angles whose sum is 90 degrees, supplementary angles whose sum is 180 degrees, and vertical angles formed by two intersecting lines which are always congruent according to the Vertical Angle Theorem. It provides examples of each kind of special angle pair and introduces the Vertical Angle Theorem stating that vertical angles are congruent.
Section1 2b
Groups of 2-3 students should draw various geometric figures involving lines and planes intersecting each other on a whiteboard. The figures include: a line intersecting a plane at a point; a vertical line intersecting both a horizontal plane and another line at a point; a horizontal plane intersecting a vertical plane along a line and containing another line; and a horizontal line intersecting a plane at a point, with the plane containing a vertical line.
Section1.3b
The document introduces the Segment Addition Postulate, which states that if point B is between points A and C, then the sum of the lengths of segments AB and BC equals the length of segment AC. It then provides an example of using this postulate to solve for the unknown value x in a geometry problem where one segment bisects another.
Section1.5
This document contains 7 statements about geometry concepts related to points, lines, and planes. Students are asked to classify each statement as True or False. The statements cover topics such as the number of planes a triangle can lie in, properties of points and lines, how many points define a plane, and how many planes can pass through given lines and points.
Section2.4
The document discusses three kinds of special angle pairs: complementary angles whose sum is 90 degrees, supplementary angles whose sum is 180 degrees, and vertical angles formed by two intersecting lines which are always congruent according to the Vertical Angle Theorem. It provides examples of each kind of special angle pair and introduces the Vertical Angle Theorem stating that vertical angles are congruent.
Section1.2b
Students should work in groups of 2-3 to draw geometric figures involving lines and planes intersecting each other. They should draw the figures requested, label everything clearly, and present their answers on the board. The figures include: a line intersecting a plane at a point; a vertical line intersecting a horizontal plane and another line at a point; a horizontal plane intersecting a vertical plane along a line and containing another line; and a horizontal line intersecting a plane at a point, with the plane containing a vertical line.
Section1.2a
The document defines the basic terms in geometry:
1) A point has 0 dimensions and represents a location in space.
2) A line extends in one direction and has 1 dimension (length).
3) A plane extends indefinitely in two directions and has 2 dimensions (length and width).
It also defines related terms like space, collinear points, coplanar points, and intersections. The document concludes with practice problems classifying statements as true or false.
Section1.3a
This document defines several key geometry terms:
- A segment is defined as two points on a line connected by all points between them. The endpoints are the defining points of the segment.
- A ray is one endpoint point on a line extending in one direction. Opposite rays extend from the same endpoint in opposite directions.
- The length of a segment is the distance between its two endpoints. On a number line, length is the absolute value of the difference between the coordinates.
- Other terms defined include postulates, congruent, congruent segments, midpoint of a segment, and segment bisector. Practice problems are provided to reinforce the new definitions.
Section1.4
The document introduces several key concepts about angles:
1) It defines an angle as the union of two rays with a common endpoint called the vertex.
2) It classifies angles as acute, right, or obtuse based on their measure.
3) It defines congruent angles, adjacent angles, and a linear pair of angles.
4) It introduces the angle bisector, angle addition postulate, and linear pair postulate as tools for working with angles.
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Section1 3a
- 1. Segments, Rays, & Distance Section 1-3
- 2. And Now, for Some New Definitions Segment – two points on a line and all the points between them B C Segment BC, or… BC B and C are the endpoints of BC. In geometry, between means “on the line connecting.”
- 3. More New Definitions Ray – one point on a line and all the points extending in one direction only C K Ray CK, or… CK (not KC!!!) C is the endpoint of CK.
- 4. More New Definitions Opposite rays – rays that extend in opposite directions along the same line from a common endpoint X Y Z YX and YZ are opposite rays.
- 5. More New Definitions Length – the distance from one point to another X W Y Z • • • • -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 The length of XY is denoted XY. To find lengths on a number line, take the absolute value of the difference of the coordinates of the endpts.: XY = 3−(−5) =8 OR XY = −5−3 =8 Distance must be positive.
- 6. More New Definitions Postulates – statements accepted without proof; also called axioms Congruent – having the same size and shape congruence symbol:
- 7. More New Definitions Congruent Segments – segments of equal length •N 10 cm P• 10 cm •M L• To show congruence in the diagram The segments have Which means that they equal lengths: are congruent: LN = MP LN = MP
- 8. More New Definitions Midpoint of a Segment – point that divides a segment into two congruent segments • • • A B C If AB = BC, then B is the midpoint of AC.
- 9. One More Definition Segment Bisector – a line, segment, ray, or plane that intersects a segment at its midpoint
- 10. Let’s Practice!!! Look to your textbooks, page 14, #1-14