7th Grade Chapter 9
9-1 Introduction to Geometry: Points, Lines, and Planes
Point, line and plane are geometric concepts. A point has no dimensions. A line is one-dimensional with no width or height. You cannot measure length of a line as it has no end. A plane has two dimensions, length and width, but no height. Length and width of a plane extend forever. The simplest figure you can measure is a line segment. The terms intersecting, parallel, and skew describe the relationship between lines. Intersecting lines have exactly one point in common. Parallel lines lie in the same plane, while skew lines lie in different planes, they have no points in common.
9-2 Angle Relationships and Parallel Lines
Special relationships exist between some angles. Identifying these special relationships can be useful when solving everyday problems in many areas, including art, nature, and architecture.
9-3 Classifying Polygons
The prefixes, tri-, means three, penta-, meaning five, hexa-, meaning six, and octo-, meaning eight, come from the Greek language. (The prefix quad-, meaning four, comes from the Latin language.) The term polygon has Greek roots as well. Polygon is based on the Greek words polus, meaning “many”, and gonia, meaning “angle.”
9-4 Reasoning Strategy: Draw a Diagram
Drawing a diagram often helps you model relationships or understand a situation more clearly. A diagram can also serve as a written record of your thought processes and lets you see the steps you have taken to solve a problem. By analyzing the data and organizing it into your diagram, you may see patterns, or even an algebraic equation, to use in solving the problem. Finally, a diagram can help you explain your reasoning.
9-5 Congruence
Congruent polygons have the same size and shape. However, congruent polygons are not considered “equal.” The measures of corresponding parts of congruent polygons are equal.
9-6 Circles
The number pi, which equals the ratio of a circle’s circumference to its diameter, is irrational. It cannot be written as the ratio of two integers. You may use 22/7 or 3.14 as an approximation of pi. Results of calculations using any approximation of pi, including pi on a calculator, should be expressed using the word or sign ‘about’.
9-7 Constructions
When you do geometric constructions, you draft a figure using only two tools, a compass and a straightedge (which cannot be using for measuring). Euclid stated these limitations in his book, Elements, about 300 B.C. With the compass you can compare two lengths, though you cannot measure them in units. Do constructions carefully. Use a finely-sharpened pencil to find points of intersection accurately.
9-8 Translations
Transformation geometry is the study of rigid motions of a figure in a plane. Rigid motions (translations, rotations, and reflections) preserve the shape and size (congruency) of a figure while changing its location.
9-9 Symmetry and Reflections
You can think of a line of symmetry as a line along which you can fold a figure in half so that both halves match at every point. You can fold along a line of reflection and the figure and its image will coincide. An image and its reflection are congruent.
9-10 Rotations
A figure has rotational symmetry if you can turn it about a point and, at some place prior to a complete turn, the turning figure matches the original figure. The figure and its rotation image are congruent.
Point, line and plane are geometric concepts. A point has no dimensions. A line is one-dimensional with no width or height. You cannot measure length of a line as it has no end. A plane has two dimensions, length and width, but no height. Length and width of a plane extend forever. The simplest figure you can measure is a line segment. The terms intersecting, parallel, and skew describe the relationship between lines. Intersecting lines have exactly one point in common. Parallel lines lie in the same plane, while skew lines lie in different planes, they have no points in common.
9-2 Angle Relationships and Parallel Lines
Special relationships exist between some angles. Identifying these special relationships can be useful when solving everyday problems in many areas, including art, nature, and architecture.
9-3 Classifying Polygons
The prefixes, tri-, means three, penta-, meaning five, hexa-, meaning six, and octo-, meaning eight, come from the Greek language. (The prefix quad-, meaning four, comes from the Latin language.) The term polygon has Greek roots as well. Polygon is based on the Greek words polus, meaning “many”, and gonia, meaning “angle.”
9-4 Reasoning Strategy: Draw a Diagram
Drawing a diagram often helps you model relationships or understand a situation more clearly. A diagram can also serve as a written record of your thought processes and lets you see the steps you have taken to solve a problem. By analyzing the data and organizing it into your diagram, you may see patterns, or even an algebraic equation, to use in solving the problem. Finally, a diagram can help you explain your reasoning.
9-5 Congruence
Congruent polygons have the same size and shape. However, congruent polygons are not considered “equal.” The measures of corresponding parts of congruent polygons are equal.
9-6 Circles
The number pi, which equals the ratio of a circle’s circumference to its diameter, is irrational. It cannot be written as the ratio of two integers. You may use 22/7 or 3.14 as an approximation of pi. Results of calculations using any approximation of pi, including pi on a calculator, should be expressed using the word or sign ‘about’.
9-7 Constructions
When you do geometric constructions, you draft a figure using only two tools, a compass and a straightedge (which cannot be using for measuring). Euclid stated these limitations in his book, Elements, about 300 B.C. With the compass you can compare two lengths, though you cannot measure them in units. Do constructions carefully. Use a finely-sharpened pencil to find points of intersection accurately.
9-8 Translations
Transformation geometry is the study of rigid motions of a figure in a plane. Rigid motions (translations, rotations, and reflections) preserve the shape and size (congruency) of a figure while changing its location.
9-9 Symmetry and Reflections
You can think of a line of symmetry as a line along which you can fold a figure in half so that both halves match at every point. You can fold along a line of reflection and the figure and its image will coincide. An image and its reflection are congruent.
9-10 Rotations
A figure has rotational symmetry if you can turn it about a point and, at some place prior to a complete turn, the turning figure matches the original figure. The figure and its rotation image are congruent.